Outline of Quantum Mechanics - · PDF fileOUTLINE OF QUANTUM MECHANICS 3 The mathematical...

Contemporary Mathematics

Outline of Quantum Mechanics

William G. Faris

Abstract. This is a brief outline of the mathematics of quantum mechanics.It begins with examples of unitary time evolution given by the Schrodingerequation. It is then shown how the spectral theorem for self-adjoint operatorsgives a general framework for studying solutions of the Schrodinger equation.There is a discussion of the role of Planck’s constant and uncertainty principles.This is followed by a section on spin and statistics. The exposition concludeswith remarks about the various roles played by self-adjoint operators in theformulation of quantum mechanics. While the main purpose of this outline isto give a succinct and mathematically correct account of the basic notions ofquantum mechanics, there is also an attempt to explain why giving a coherentinterpretation of quantum mechanics is so difficult.

Contents

1. The setting for quantum mechanics 21.1. Introduction 21.2. Plan of the exposition 41.3. Hilbert space 51.4. Unitary operators 62. The Schrodinger equation 82.1. Diffusion and the free motion Schrodinger equation 82.2. The free particle 92.3. Shift operators 122.4. The Schrodinger equation with a potential energy function 122.5. Motion with constant force 132.6. Spectral and propagator solutions 142.7. Particle in a box 152.8. Particle on the half-line 172.9. The diffusion with drift representation 182.10. The harmonic oscillator 19

1991 Mathematics Subject Classification. 81-01.Key words and phrases. quantum mechanics, quantum state, wave function, Schrodinger

equation, self-adjoint operator, spectral representation, Dirac notation, unitary dynamics, quan-tum measurement.

c© 2010 by the author. This paper may be reproduced, in its entirety, for non-commercialpurposes.

c©0000 (copyright holder)

1

2 WILLIAM G. FARIS

3. Self-adjoint operators 233.1. Stone’s theorem 233.2. Multiplication operators 243.3. The spectral theorem 253.4. Spectral measures 263.5. Generalized vectors 283.6. Dirac notation 293.7. Spectral representation for multiplicity one 313.8. Form sums and Schrodinger operators 334. The role of Planck’s constant 354.1. The uncertainty principle 354.2. Classical mechanics 375. Spin and statistics 385.1. Spin 1

2 385.2. Composite systems 405.3. Statistics 426. Fundamental structures of quantum mechanics 426.1. Self-adjoint operators as dynamics 426.2. Self-adjoint operators as states 436.3. Self-adjoint operators as observables? 466.4. Self-adjoint operators with measurement as observables 48Acknowledgments 52References 52

1. The setting for quantum mechanics

1.1. Introduction. Quantum mechanics is the framework for the current the-ory of matter, viewed as consisting of constituents on the molecular, atomic, andsub-atomic scale. The characteristic feature of quantum mechanics is the occurrenceof Planck’s constant

(1.1) ~ = 1.05443× 10−27erg s.

This has the dimensions of energy times time, or momentum times distance. Inother words, it has the dimensions of angular momentum. This number is unbe-lievably small, but it affects many features of our world. The mass of an electronis

(1.2) m = 9.1083× 10−28g.

The ratio

(1.3) σ2 =~

m= 1.157cm2/s

has the dimensions of a diffusion constant. This is the parameter that determinesthe propagation of a free electron. It is perhaps too optimistic to think of quantummechanics as a theory that can be derived from classical mechanics by some processcalled quantization. It may even be misleading to think of classical mechanics asa limit of quantum mechanics, in view of the fact that ~ is a fixed number. Onthe other hand, there are various scales of nature on which quantum diffusion isnegligible, and on these scales the object should behave like its classical counterpart.

OUTLINE OF QUANTUM MECHANICS 3

The mathematical structure of quantum mechanics is based on three funda-mental ideas. These are:

• State• Time evolution• Observable

As we shall see, the first two may be defined as mathematical objects: vectorsin Hilbert space and unitary operators acting on these vectors. The notion ofobservable will turn out to be remarkably obscure. In some accounts it is identifiedwith the notion of self-adjoint operator, but we shall see that this leads to problems.

The mathematical description of a quantum state is as a unit vector in a Hilbertspace. Often this is represented in a concrete form as a normalized wave function,that is, a complex valued function depending on a space variable. The normalizationcondition says that the absolute value squared of this function is a probabilitydensity, that is, has integral equal to one. In this case it is regarded as giving theprobability density describing the random position of a particle at a single fixedtime.

Time evolution in quantum mechanics is given by unitary linear operatorsfrom the Hilbert space to itself. Each such operator is simply an automorphismof the Hilbert space. Typically, these unitary operators fit together to form aone-parameter group of unitary operators, where the parameter is the time dis-placement. In practice this time evolution is obtained by solving a conservativeevolution equation, the famous Schrodinger equation. It is an odd situation: thedynamics takes place on the level of the wave function, but the observed quantityis interpreted as a particle.

An observable is sometimes identified as a self-adjoint operator acting in theHilbert space. It behaves something like a random variable in probability theory,but there is a crucial difference. Given a quantum state, a self-adjoint operatorgives rise to a random variable. More generally, a family of commuting self-adjointoperators gives rise to a family of random variables on the same probability space.There is a problem, however, with interpreting the values of these random variablesas describing intrinsic properties of the system. The problem comes when oneconsiders non-commuting families of self-adjoint operators. As we shall see, thereis a serious consistency problem that prevents such an interpretation.

A possible resolution of this issue is to maintain that different observables cor-respond to different experimental contexts. There is no absolute notion of the valueof an observable, apart from such a context. In particular, a measurement of oneobservable may preclude the measurement of another observable. However, if thenotion of measurement is needed to complete the concept of observable, then thereis the problem of describing exactly what constitutes a measurement. It is thusdoubtful that every self-adjoint operator represents an observable in the sense de-scribed here. In fact, it may be that the question of which self-adjoint operators areassociated with experimental measurements and thus represent observables dependson the details of the time evolution.

It is generally regarded that many of the most fundamental self-adjoint opera-tors correspond to observables. These include operators corresponding to position,momentum, energy, and spin component. Some physicists have maintained thatthese all reduce to macroscopic position measurements (for example pointer read-ings) via some sort of measurement process. For example, when a particle with

4 WILLIAM G. FARIS

spin is passed through a region where there is a suitable magnetic field, then itis deflected in one way or the other, and this deflection is amplified into a macro-scopic difference in position. Thus the apparatus with the magnetic field produces ameasurement of the spin component along the field direction. This does not meanthat the value of the spin component exists before the measurement; it emergesfrom the interaction with a particular apparatus, and it is this apparatus that de-fines the particular component being measured. The measurement is ultimately adetermination of the position of the particle after the deflection.

Even position is problematical in quantum mechanics. The theory providesno mechanism for defining particle trajectories. The wave function determines theprobability density for particle position at a fixed time. It does not determine thejoint probability density for the positions of the particle at two or more differenttimes. Thus quantum mechanics involves waves and particles, but their roles arevery different. The waves propagate according to a well-defined linear partial dif-ferential equation. There is no description of particle motion, but it is the particlesthat are ultimately observed.

1.2. Plan of the exposition. The following account begins with an accountof states as vectors in Hilbert space and of time evolutions for particular systems.In other words, the story is about the Schrodinger equation.

The development then proceeds to the spectral theorem for self-adjoint op-erators. This theorem has remarkable consequences. It emerges that the self-adjoint operators on a Hilbert space are in one-to-one correspondence with theone-parameter groups of unitary operators. In particular, there is a self-adjointoperator that determines the dynamics given by the Schrodinger equation.

The next parts are about special topics. Planck’s constant and the uncertaintyprinciple have important physical consequences. In many circumstances when thepotential energy has negative singularities, the total energy is bounded below, witha bound that depends on the value of Planck’s constant. This is at least part of theexplanation for the stability of atoms. Other topics include spin and statistics. Thefact that electrons obey Fermi-Dirac statistics leads to the Pauli exclusion principle,another ingredient in explaining the stability of matter.

Finally, there is a discussion of other roles for self-adjoint operators. Anotherconsequence of the spectral theorem is that, given a particular quantum state, everyself-adjoint operator may be realized as a random variable. This tempts some tosay that every self-adjoint operator is an observable, and certainly there is nothingto forbid this choice of terminology. However if the notion of observable is restrictedto something that can be measured in an experiment, then it most likely the casethat relatively few self-adjoint operators correspond to observables. For the onesthat do correspond to observables it is necessary to give an account of how this isaccomplished by measurement. A complete account is not known to the author,but the final section will give some hints suggested by the conventional wisdom ofquantum mechanics. The perceptive reader will notice the occurrence of the word“macroscopic,” which is a hint that nothing is understood.

It should be emphasized that there is very little that is controversial aboutquantum dynamics. The part that is obscure is how the objects described by thedynamics are reflected in reality.


Before proceeding to the topics outlined in this subsection, there is one morepoint that should be mentioned. The usual expositions of quantum theory, includ-ing this one, emphasize the vector space description of quantum states. However,while in quantum mechanics each non-zero vector determines a state, it turns outthat two vectors that are multiples of each other determine the same state. Asa consequence, that space of states in quantum mechanics is actually a projectivespace. Furthermore, while the automorphisms of this space are typically determinedby unitary operators, in some circumstances anti-unitary operators are appropriate.While these are important foundational issues, they will not play a major role inwhat follows. See the appendix to David Wick’s book [12] for further informationabout the projective space interpretation.

Here are warnings about terminology and notation. In the following, whenthere is talk of real or complex functions on some set, it is taken that the sethas the structure of a standard measurable space and that the functions are Borelmeasurable functions. In general, the goal is to bypass technical issues that are notrelevant to the discussion of the main subject. For instances, Hilbert spaces arealways taken to be separable. For simplicity almost all of what follows will be aboutquantum mechanics in one space dimension. Nevertheless, space derivatives arewritten in the partial derivative notation ∂

∂x. Such space derivatives are interpreted

in a non-classical sense: ∂∂xψ = χ holds whenever ψ is an indefinite integral of the

locally integrable function χ. (That is, ψ is required to be an absolutely continuousfunction.) In particular, ψ can have a slope discontinuity. The term operator is asynonym for linear transformation defined on a vector space. For instance, in thefollowing ∂

∂xis typically interpreted as an operator defined on the vector space of

all square-integrable ψ such that ∂∂xψ = χ is also square-integrable. Since every

square-integrable function χ on the line is also locally integrable, the derivativemakes sense.

1.3. Hilbert space. A Hilbert space H (or complex Hilbert space) is a com-plex vector space with an inner product that is a complete metric space with respectto the norm associated with the inner product. In general, we shall write a vectorin H as ψ and its norm as ‖ψ‖.

The definition of inner product that we use is the following. For each orderedpair of vectors φ and ψ in H, the inner product is a complex number 〈φ, ψ〉. Theinner product must have the following properties.

(1) Linear in the second variable: 〈φ, aψ + bχ〉 = a〈φ, ψ〉 + b〈φ, χ〉.(2) Conjugate linear in the first variable: 〈aψ + bχ, φ〉 = a〈ψ, φ〉 + b〈χ, φ〉.(3) Hermitian symmetric: 〈φ, ψ〉 = 〈ψ, φ〉.(4) Positive on all vectors: 〈ψ, ψ〉 ≥ 0.(5) Strictly positive on non-zero vectors: If ψ 6= 0, then 〈ψ, ψ〉 > 0.

The convention that the inner product is linear in the second variable is some-what unusual in the mathematics literature, but it is quite standard in physics. Oneadvantage is that it gives a particularly convenient notation for the dual space, thatis, the space of continuous linear functions from the Hilbert space to the complexscalars. Thus, if φ is a vector in H, then φ∗ is the element of the dual space suchthat the value of φ∗ on ψ is 〈φ, ψ〉. A fundamental result of Hilbert space theorystates that every element of the dual space may be represented in this way. Thisnotation is consistent with the usual notation in matrix theory, in which ψ is given

6 WILLIAM G. FARIS

by a column vector, and φ∗ is a row vector. The inner product φ∗ψ is a scalar,while the outer product ψφ∗ is a linear transformation from the Hilbert space toitself (a projection). In particular, the inner product ψ∗ψ = ‖ψ‖2, while ψψ∗ is anorthogonal projection.

The norm associated with the inner product is ‖ψ‖ =√

〈ψ, ψ〉. The positivityaxiom says that ‖ψ‖ ≥ 0; the strict positivity says that for ψ not equal to thezero vector we even have ‖ψ‖ > 0. It is assumed that the reader is familiar withthe properties of inner products and their associated norms. As usual, we say thatφ and ψ are orthogonal if 〈φ, ψ〉 = 0. In this case we write φ⊥ψ, and we have‖φ‖2 + ‖ψ‖2 = ‖φ+ ψ‖2. This is the theorem of Pythagoras.

A norm defines a metric that makes the space into a metric space. To say thatthe Hilbert space is a complete metric space is to say that every Cauchy sequenceof vectors in the Hilbert space converges to a vector in the Hilbert space.

Here is one of the most useful examples of a Hilbert space. Consider the spaceH of complex functions ψ defined on the real line such that

(1.4) ‖ψ‖2 =

∫ ∞

−∞

|ψ(x)|2 dx <∞.

Define the inner product by

(1.5) 〈φ, ψ〉 =

∫ ∞

−∞

φ(x)ψ(x) dx <∞.

This satisfies all the axioms, except for the strict positivity. However if we regardeach pair of functions that differ on a set of measure zero as defining the same ele-ment of the Hilbert space, then after this identification all the axioms are satisfied.The resulting Hilbert space is denoted H = L2(R, dx). This notation indicates thatthe functions are complex functions on the real line R that are square integrablein the sense of Lebesgue. In the quantum mechanical context, such a function iscalled a wave function.

Another important example of a Hilbert space is an ℓ2 space of complex se-quences that are square-summable with respect to a given weight function. Theweights wj > 0 are strictly positive. A sequence c in this space satisfies

(1.6) ‖c‖2 =∑

j

wj |cj |2 <∞.

In quantum mechanics there is always an underlying Hilbert space H. A stateis determined by a unit vector ψ in H. Two unit vectors define the same state ifone is a complex multiple of the other. Since the two vectors have the same length,this complex multiple must be a phase eiθ.

1.4. Unitary operators. A Hilbert space isomorphism from a complex Hilbertspace H to another complex Hilbert space H′ is a linear transformation U : H → H′

that is a bijection and also preserves the norm, so that ‖Uψ‖ = ‖ψ‖. It may beshown that a Hilbert space isomorphism automatically preserves the inner product.

Here is a fundamental example, the Fourier transform. The Fourier transform

of the complex function ψ(x) in H = L2(R, dx) is another complex function ψ(k).

The mapping F that takes ψ to ψ is the Fourier transform mapping. It is a unitary


operator from the Hilbert space H to the Hilbert space H = L2(R, dk2π ) with norm

(1.7) ‖ψ‖2 =

∫ ∞

−∞

|ψ(k)|2 dk2π.

It is determined by the relation

(1.8) ψ(k) =

∫ ∞

−∞

e−ikxψ(x) dx.

The identity that expresses the fact that F is unitary from H to H is

(1.9)

∫ ∞

−∞

|ψ(k)|2 dk2π

=

∫ ∞

−∞

|ψ(x)|2 dx.

We can write this as

(1.10) ψ = Fψ,

with

(1.11) ‖ψ‖2 = ‖ψ‖2.

Notice that the integral over k that defines the left hand side has an extra factorof 1/(2π). The inverse F−1 of the Fourier transform is a unitary operator from Hto H determined by

(1.12) ψ(x) =

∫ ∞

−∞

eikxψ(k)dk

2π.

The Fourier transform variable k is called the wave number, and its dimension isthat of inverse length. Thus to the extent that we can compute these integrals,we get a way of translating statements about position x to statements about wavenumber k. Sometimes a Hilbert space isomorphism is called a unitary operator. Inthe following we shall mainly refer to a unitary operator as a linear transformationU : H → H of a complex Hilbert space to itself that is a Hilbert space isomorphism.

We shall often use the fact that the Fourier transform of −i ∂∂xψ(x) is kψ(k)

and that the Fourier transform of xψ(x) is i ∂∂kψ(k). In fact, for many purposes we

could think of these as defining the derivatives.Here is a fundamental example of a unitary operator. Consider the space

H = L2(R, dk2π ). Let α(k) be a real function on the real line. Then the operator Vdefined by

(1.13) (V ψ)(k) = e−iα(k)ψ(k)

is a unitary operator. A composition of unitary operators is again a unitary oper-ator. For example, the composition U = F−1V F is a quite non-trivial example ofa unitary operator.

Often unitary operators combine to form a one-parameter unitary group. Thisrefers to a homomorphism of the additive group of the line to the group of unitaryoperators on the Hilbert space H. In other words, for each real t there is a unitaryoperator Ut : H → H. The group homomorphism property says that U0 = I andthat

(1.14) Ut+t′ = UtUt′ .

Finally, it is require that for each ψ in H the map t 7→ Utψ is continuous from thereal line to H.

8 WILLIAM G. FARIS

Here is an example of such a unitary group. Consider the space H = L2(R, dk2π ).Let α(k) be a real function on the real line. Then for each t the operator Vt definedby

(1.15) (Vtψ)(k) = e−itα(k)ψ(k)

is a unitary operator, and this defines a unitary group. The composition Ut =F−1VtF provides a non-trivial example of a unitary group.

There are analogous constructions for isomorphisms with ℓ2 spaces. Say that φjis an orthogonal basis for the Hilbert space H. Define the ℓ2 space with the weightfunction wj = 1/〈φj , φj〉. Then the map Z : H → ℓ2 given by (Zψ)j = 〈φj , ψ〉 is anisomorphism from H to ℓ2. The inverse isomorphism is given by Z−1c =

∑

j wjcjφj .Suppose that λj are real numbers. Then the composition given by Z followed bymultiplication by e−itλj followed by Z−1 defines a unitary group Ut. Explicitly

(1.16) Utψ =∑

j

wjφje−itλj 〈φj , ψ〉.

2. The Schrodinger equation

2.1. Diffusion and the free motion Schrodinger equation. The diffusionequation (or heat equation) describes diffusing particles. The equation is

(2.1)∂

∂tu =

σ2

2

∂2

∂x2u

We look for a solution u(x, t) = (Qtψ)(x) satisfying u(x, 0) = ψ(x). In order toapply the Fourier transform, we take ψ in L2(R). The Fourier transform satisfies

(2.2)d

dtu = −σ

2

2k2u

with initial condition u(k, 0) = ψ(k). This has solution

(2.3) u(k, t) = e−σ2

2 k2tψ(k).

Recall that the Gaussian density with variance parameter ǫ2 is defined by

(2.4) gǫ2(x) =1√2πǫ2

e−x2

2ǫ2 .

Its Fourier transform is

(2.5) gǫ2(k) =

∫ ∞

−∞

eikxgǫ2(x) dx = e−ǫ2k2

2

The expression for u(k, t) is the product of the Fourier transform of the Gauss-

ian density of variance σ2t with ψ(k). To evaluate the inverse Fourier transformu(x, t) use the fact that the inverse Fourier transform of a product is a convolution.This proves the following result.

Proposition 2.1. The solution of the diffusion equation is given in terms ofthe Fourier transform F for t ≥ 0 by

(2.6) u(t) = Qtψ = F−1QtFψ,


where Qt is multiplication by e−σ2tk2

2 . Its explicit form for t > 0 is given by theconvolution

(2.7) u(x, t) = (Qtψ)(x) = (gσ2t ∗ ψ)(x) =

∫ ∞

−∞

1√2πσ2t

e−(x−y)2

2σ2t ψ(y) dy.

Notice that Qt is not unitary, but it does satisfy the stability condition ‖Qtψ‖ ≤‖ψ‖ for t > 0. Typically the initial condition ψ(x) is taken to be a positive function,and then it follows that the solution u(x, t) ≥ 0 is also positive. Furthermore, inthis case the total integral is constant. This explains why the solution is interpretedas a density.

The interpretation of the Schrodinger equation is quite different. The functionψ represents the quantum state. In this case the position observable is the xcoordinate. If ψ is normalized so that the L2 norm ‖ψ‖ = 1, then |ψ(x)|2 is aprobability density for x. The state makes the observable a random variable: theexpectation of a function f(x) is given by

(2.8) Eψ [f(x)] =

∫ ∞

−∞

f(x)|ψ(x)|2 dx.

The Schrodinger equation for free motion is

(2.9)∂

∂tu = i

σ2

2

∂2

∂x2u.

The same manipulations as for the diffusion equation give the solution

(2.10) u(k, t) = e−iσ2

2 k2tψ(k).

Proposition 2.2. The solution of the free Schrodinger equation is given interms of the Fourier transform F for real t by

(2.11) u(t) = Utψ = F−1UtFψ,

where Ut is multiplication by e−iσ2tk2

2 . The solution operator Ut is unitary. Itsexplicit form for t 6= 0 is given by the convolution

(2.12) u(x, t) = (Utψ)(x) = (giσ2t ∗ ψ)(x) =

∫ ∞

−∞

1√2πiσ2t

ei(x−y)2

2σ2t ψ(y) dy.

The only modification that must be made to pass from the diffusion equationto quantum mechanics is to replace t by it. The same kinds of formula continue tohold true, even though the convergence of the integrals is more delicate.

2.2. The free particle. The Schrodinger equation for free motion introducedin the previous section defines a simple but non-trivial conservative dynamics thatdepends only on a diffusion constant. It is conservative in the sense that the solutionoperator is unitary and in particular preserves the L2 norm. The relevance of thisdynamics to quantum physics is an empirical fact that cannot be derived solelyfrom mathematics. However, it gives predictions that are not difficult to interpret.In particular, the free particle solution describes motion at constant velocity, butwith a spread of initial velocities. Here is a more detailed description of how thisworks.

10 WILLIAM G. FARIS

Both the spatial shift and the time dynamics have nice expressions in terms ofwave number. The spatial shift is a unitary operator from H to H with the obviousexpression

(2.13) (Vaψ)(x) = ψ(x− a).

This may be written

(2.14) Va = F−1VaF,

where the unitary operator Va from H to H is given by

(2.15) (Vaψ)(k) = e−iakψ(k).

Notice that a has the dimensions of length, while k has the dimensions of inverselength, so ak is dimensionless, as it must be.

The free particle unitary time dynamics Ut on H is given by a similar expression.Now we have t with the dimension of time, and k with the dimension of inverselength. In quantum mechanics we have the diffusion constant 1

2σ2 with dimensions

of length squared over time. So a suitable dimensionless combination is t 12σ2k2.

The corresponding unitary operator on H is

(2.16) (Utψ)(k) = e−it12σ

2k2

ψ(k).

In the position representation we have

(2.17) Ut = F−1UtF.

We have already computed this explicitly as a convolution by a Gaussian witha complex variance parameter. The result of this explicit computation has aninteresting physical interpretation.

Proposition 2.3. Define the unitary operator from H to H by

(2.18) (Mtψ)(y) = eiy2

2σ2tψ(y).

Define another unitary operator from H to H by

(2.19) (Ztχ)(x) =1√

2πiσ2tχ(

x

σ2t).

Then the free particle solution operator is given by

(2.20) Ut = MtZtFMt.

This identity says that the solution Utψ = MtZtFMtψ is obtained by multiply-ing by a complex phase, taking the Fourier transform, rescaling, and multiplyingagain by a complex phase. It may be derived from the explicit solution (Utψ)(x)of the Schrodinger equation as convolution by a complex Gaussian. It is sufficientto expand the quadratic expression in the exponent as the sum of three terms andwrite the result as the product of three exponentials. The result is (MtZtFMtψ)(x).

This last identity determines the long time asymptotics of the solution. Con-sider the difference Utψ −MtZtFψ. This has norm

(2.21) ‖Utψ −MtZtFψ‖ = ‖MtZtFMtψ −MtZtFψ‖.This is equal to

(2.22) ‖MtZtF (Mtψ − ψ)‖ = ‖Mtψ − ψ‖.


However the right hand side goes to zero as t → ±∞. This proves the followingresult.

Proposition 2.4. In the limit t→ ±∞ the solution of the free particle Schrodingerequation has the asymptotic form

(2.23) Utψ ∼MtZtψ.

Explicitly,

(2.24) (Utψ)(x) ∼ eix2

2σ2t

1√2πiσ2t

ψ(x

σ2t).

This identity says that the asymptotic solution MtZtFψ = MtZtψ is obtainedby taking the Fourier transform, rescaling, and multiplying by a complex phase.Once the Fourier transform has been performed, the remaining operations are sim-ple and explicit. The main effect is the rescaling. This leads to a velocity ofpropagation at wave number k that is asymptotically σ2k. When t is large and x

is large with x/t ∼ σ2k, the amplitude (Utψ)(x) is proportional to the value ψ(k).The solution spreads out in space according to wave number. This next propositiongives a particularly simple form for the asymptotic position probability density.

Proposition 2.5. The position probability density for the free particle is givenasymptotically by

(2.25) |(Utψ)(x)|2 ∼ 1

2πσ2t|ψ(

x

σ2t)|2.

This leads to the notion of taking the wave number k or the corresponding

velocity σ2k as an observable . Its probability density would then be |ψ(k)|2. Theexpectation of a function of k is given by

(2.26) Eψ[f(k)] =

∫ ∞

−∞

f(k)|ψ(k)|2 dk2π

=

∫ ∞

−∞

f(x

σ2t)

1

2πσ2t|ψ(

x

σ2t)|2 dx.

Compare this to

(2.27) EUtψ[f(x2

σ2t)] =

∫ ∞

−∞

f(x

σ2t)|(Utψ)(x)|2 dx

The asymptotic relation expressed in the previous proposition leads to the followingimportant result.

Proposition 2.6. An expectation involving wave number may be expressed interms of asymptotic position by

(2.28) Eψ[f(k)] ∼ EUtψ[f(

x

σ2t)].

In quantum mechanics it is customary to take the observable as the momentump = ~k = mσ2k instead of the velocity σ2k. So the notion is that position andmomentum are both observables, but a measurement of one is incompatible withthe measurement of the other. This is because for position one uses the positiondensity |ψ(x)|2 now, while for momentum one uses the position density |(Utψ)(x)|2much later, that is, for large t.

If the probability for x is initially very localized near a point, then there will bemany wave numbers represented in the Fourier transform, and so the probabilityfor k will be spread out. In particular, there will be many different propagation

12 WILLIAM G. FARIS

velocities, and with time the dynamics will filter the various wave numbers to makethem occupy different regions in space.

2.3. Shift operators. In quantum mechanics there is a remarkable dualitybetween position and wave number (or momentum). The position shift operatorVa is the unitary operator given by

(2.29) (Vaψ)(x) = ψ(x− a).

We have already seen its form in the wave number representation. The wave number

shift operator Wb is the unitary operator given by

(2.30) (Wbψ)(k) = ψ(k − b).

Since the momentum is p = ~k, this is closely related to the corresponding momen-tum shift operator. In the position representation this shift operator becomes

(2.31) (Wbψ)(x) = eibxψ(x).

These unitary operators satisfy the algebraic relation

(2.32) WbVa = eiabVaWb.

That is, if the order of the shifts are reversed, then the result is the same, up toa phase factor. Of course this phase factor makes no difference when determiningthe quantum state.

There is a remarkable uniqueness theorem that says that there is only oneirreducible solution of this operator equation, up to isomorphism. This theoremgives an abstract characterization of the framework involving position and wavenumber and the Fourier transform. This shows that the basic structure of quantummechanics may be derived from a purely algebraic relation for unitary operators.

2.4. The Schrodinger equation with a potential energy function.

Schrodinger discovered the fundamental dynamical equation of quantum mechanics.The most common way of writing the Schrodinger equation is

(2.33) i~∂

∂tu = − ~

2

2m

∂2

∂x2u+ v(x)u.

Both sides of the equation have the units of energy. Here v(x) is the potentialenergy function. The problem is to find the solution u(x, t) of this equation withsome initial condition u(x, 0) = φ(0).

For some purposes it is convenient to write the equation in a slightly differentbut equivalent form. The only changes are the sign change and division by ~. Thisother form is

(2.34) −i ∂∂tu =

1

2σ2 ∂

2

∂x2u− 1

~v(x)u.

Both sides of this equation have the units of inverse time. The potential energyterm is divided by ~ in order to convert energy units into inverse time units. Foreach form of the potential energy function v(x) one has a separate problem, andsuch problems are difficult to solve in general. We shall look at a few simple caseswhere it is possible to make progress.

There is a closely related equation diffusion with removal equation

(2.35)∂

∂tu =

1

2σ2 ∂

2

∂x2u− 1

~v(x)u.


Suppose that v(x) ≥ 0. Then this equation has an interpretation in terms ofdiffusion. The σ2 is the usual diffusion constant. The v(x)/~ indicates a spatiallydependent rate at which the diffusing particles are being removed from the system.So the particles diffuse and occasionally vanish.

2.5. Motion with constant force. The next natural topic is the problemof Galileo: find the motion of a particle under the influence of a constant force ǫ.The solution would not have surprised Galileo: constant acceleration implies linearincrease in velocity and quadratic increase in position.

Proposition 2.7. The solution of the Schrodinger equation with constant forceǫ is

(2.36) u(t) = Gtψ = c(t)W ǫt~

V 12

ǫmt2Utψ.

Here c(t) is a phase factor. The unitary operator Ut describes the free particleevolution. The unitary operator W ǫt

~

represents a shift in momentum p = ~k by ǫt.

The unitary operator V 12

ǫmt2 represents a shift in position by 1

2ǫmt2.

One can write the solution in a more explicit form as

(2.37) (Gtψ)(x) = c(t)eiǫt~x(Utψ)(x − 1

2

ǫ

mt2).

The value of the phase c(t) is not important for the physics, but if we take c(t) =

e−i16

ǫ2t3

m~ , then we get the group homomorphism property Gt+t′ = GtGt′ . In anycase, it will emerge with the systematic derivation of the result.

Here is the derivation. Take v(x) = −ǫx in the Schrodinger equation. TheFourier transformed equation is then

(2.38)∂

∂tu+

ǫ

~

∂

∂ku = −i1

2σ2k2u.

This is the equation for translation in wave number space, but with a source term.This equation is not difficult to solve directly, but it is illuminating to put it in

the form of a conservation law by the change of change of variable u = a(k)w. Ifwe choose a(k) so that ǫ

~a′(k)/a(k) = −i 12σ2k2, then the new equation no longer

has a source term; it is simply the translation equation in wave number space

(2.39)∂

∂tw +

ǫ

~

∂

∂kw = 0.

.This equation is easy to solve by integrating along the lines that are solutions

of dk/dt = ǫ~. These are the lines k = k0 + ǫ

~t. Along such a line the solution is

constant, and so we have w(k, t) = f(k0) = f(k − ǫ~t). The solution is translation

at constant velocity in wave number space.We have u = a(k)w, where

(2.40) a(k) = exp(−i16

~

ǫσ2k3).

The initial conditions are related by ψ(k0) = a(k0)f(k0). So the solution for u is

(2.41) u(k, t) = a(k)w(k, t) = a(k)f(k0) = a(k)a(k0)−1ψ(k0).

14 WILLIAM G. FARIS

Insert k = k0 + ǫ~t to get

(2.42) u(k, t) = exp(−i16

σ2ǫ2

~2t3 − i

1

2

σ2ǫ

~t2k0 − i

1

2σ2k2

0t)ψ(k0).

Alternatively,

(2.43) u(k, t) = exp(−i16

ǫ2

m~t3 − i

1

2

ǫ

mt2k0 − i

1

2σ2k2

0t)ψ(k0),

where k0 = k− ǫ~t. The result stated in the proposition follows by taking the inverse

Fourier transform.

2.6. Spectral and propagator solutions. The following subsections treatother simple examples where it is possible to find explicit solutions of the Schrodingerequation with a potential energy function (or of the corresponding diffusion withremoval equation). These examples are the quantum description of a particle con-fined to a bounded interval, a particle confined to a half-line, and a particle witha linear force (the harmonic oscillator). The problem is to find the operator thatmaps the initial wave function at time zero to the wave function at time t. Typi-cally, there are two ways of exhibiting the solution. A spectral solution is obtainedby mapping the Hilbert space to a weighted L2 space of functions via a unitary op-erator W , multiplying by a complex phase e−iλ(k)t, and then mapping back to theoriginal Hilbert space via W−1. Typically, the unitary operator will be an integraloperator of the form

(2.44) (Wψ)(k) =

∫

φk(y)ψ(y) dy.

The inverse operator is of the form

(2.45) (W−1f)(x) =

∫

φk(x)f(k)w(k) dk.

in the continuous case or

(2.46) (W−1f)(x) =∑

k

φk(x)f(k)w(k)

in the discrete case. In either case the weights w(k) ≥ 0.A propagator is a family of integral operators given by a functions pt(x, y)

indexed by t. The solution is then given by

(2.47) (Utψ)(x) =

∫

pt(x, y)ψ(y) dy.

The propagator may be expanded to give a spectral solution. Sometimes thisexpansion is continuous, as in

(2.48) pt(x, y) =

∫

φk(x)e−iλ(k)tφk(y)w(k) dk.

Other times it is discrete, as in

(2.49) pt(x, y) =∑

k

φk(x)e−iλ(k)tφk(y)w(k).

Unfortunately, in most cases it is impossible to find an explicit solution of eitherkind. This means that it is necessary to use various approximation methods, suchas perturbation theory or variational calculations, in order to understand the dy-namics. These approximation methods are at the heart of the mathematics of


quantum mechanics. Unfortunately these topics are too extensive to include in anshort introductory account.

2.7. Particle in a box. Consider a particle confined to a bounded interval(often called a particle in a box). The idea is that the potential energy v(x) = 0 for0 < x < a and v(x) = +∞ elsewhere. The effect of this is to restrict the problemto the interval [0, a] and impose zero boundary conditions at the end points of theinterval when solving the Schrodinger equation.

First look for a spectral solution. There is a new phenomenon: discrete fre-quencies. This comes from solving the eigenvalue problem

(2.50) −1

2σ2 ∂

2

∂x2φn(x) = λnφn(x)

with the boundary condition φ(0) = φ(a). The solution is

(2.51) φn(x) = sin(nπx

a).

We conclude that

(2.52) λn =1

2σ2n

2π2

a2.

Multiplying by ~, we see that the discrete energy levels are

(2.53) En = ~λn =~

2

2m

n2π2

a2.

We can solve the evolution equation by writing an arbitrary initial conditionas

(2.54) ψ(x) =1

a

∞∑

n=1

cnφn(x)

Here

(2.55) cn = 2

∫ a

0

φn(y)ψ(y) dy.

Then for the diffusion equation the solution is

(2.56) u(x, t) =∑

n

e−λntcnφn(x).

This describes diffusion where the particles are killed as soon as the encounter theboundary of the interval [0, a].

For the Schrodinger equation the solution is

(2.57) u(x, t) =∑

n

e−iλntcnφn(x).

These solutions are oscillatory waves that bounce off the boundary.

Proposition 2.8. The spectral solution of the Schrodinger equation in a boxis given by a composition of unitary operators. The original Hilbert space is H =L2(0, a). It is mapped to a Hilbert space ℓ2 of sequences with norm given by

(2.58) ‖c‖2 =1

a

∞∑

n=1

|cn|2.

16 WILLIAM G. FARIS

The unitary map S from L2(0, a) to ℓ2 is given by

(2.59) (Sφ)n = 2

∫ a

0

φn(y)φ(y) dy.

The inverse map is

(2.60) (S−1c)(x) =1

a

∞∑

n=1

cnφn(x).

The solution operator for the Schrodinger equation is given by S followed by mul-tiplication by e−itλn followed by S−1.

The propagator solution gives complementary information. It may be obtainedfrom the above solution by using the Poisson summation formula. This formula

says that if f is a rapidly decreasing function with Fourier transform f , then

(2.61)m=∞∑

m=−∞

f(x− 2ma) =1

2a

n=∞∑

n=−∞

f(πn

a)e

iπnxa .

This formula is derived by noticing that the left hand side is a periodic functionwith period 2a and hence may be expanded in a Fourier series.

The previous result says that the time evolution for the diffusion equation onthe interval [0, a] is given by a propagator

(2.62) kt(x, y) =2

a

∞∑

n=1

e−λntφn(x)φn(y) =2

a

∞∑

n=1

gσ2t(πn

L) sin(

nπ

ax) sin(

nπ

ay).

Write 2 sin(nπax) sin(nπ

ay) = cos(nπ

a(x−y))−cos(nπ

a(x+y)). Convert the cosine

terms to complex exponentials and use the Poisson summation formula. The resultis that

(2.63) kt(x, y) =

∞∑

m=−∞

[gσ2t(x− y − 2ma) − gσ2t(x+ y − 2ma)].

For the Schrodinger equation we should get the correct formula if we replace tby it. This gives the following result.

Proposition 2.9. The propagator solution of the Schrodinger equation for aparticle confined to the interval [0, a] is given by the propagator

(2.64) pt(x, y) =

∞∑

m=−∞

[giσ2t(x − y − 2ma) − giσ2t(x+ y − 2ma)]

The interpretation of this result is the following. Consider x in the intervalfrom 0 to a. Take free particle solutions starting at y + 2am and take negatives offree particle solutions starting at −y + 2am′. The points y + 2am and −y + 2am′

are mirror images, where the mirror is placed at the point (m+m′)a. Thus thereis a mirror at each integer multiple of a. The final effect is a cancellation at thesemirror points that confines the wave to the interval. In fact, it is evident that thesolution vanishes when either x or y is equal to 0 or to a. There are only standingwaves that slosh back and forth in the interval.


2.8. Particle on the half-line. The particle on the half-line gives a simpleillustration of scattering. The potential energy v(x) = 0 for 0 < x < +∞ andv(x) = +∞ elsewhere. The effect of this is to restrict the problem to the interval[0,+∞) and impose zero boundary conditions at 0.

One way to get a spectral solution of the Schrodinger equation for this caseis to start with the box solution and let kn = nπ/a. Then one can let a → +∞,and the variable k becomes a continuous variable. We get a unitary map S fromL2(0,+∞) to L2(0,+∞) given by the sine transform

(2.65) (Sψ)(k) = 2

∫ ∞

0

φk(y)ψ(y) dy.

Here

(2.66) φk(x) = sin(kx).

The inverse map is also a sine transform

(2.67) (S−1c)(x) =

∫ ∞

0

φk(x)c(k)dk

π.

Proposition 2.10. Consider the quantum particle on the half-line with Hilbertspace H+ = L2(0,∞). Let S be the sine transform. Let

(2.68) λ(k) =1

2σ2k2.

Define Mt to be multiplication by e−itλ(k). Then the spectral solution for theSchrodinger equation is given by the operator composition S−1MtS.

This spectral solution gives the a formula for the propagator in the spectralform

(2.69) pt(x, y) =2

π

∫ ∞

0

e−itλ(k) sin(kx) sin(ky) dk.

This can also be written as

pt(x, y) =1

π

∫ ∞

0

e−itλ(k)[cos(x− y) − cos(x+ y)] dk

=1

2π

∫ ∞

−∞

e−i12σ

2tk2

[eik(x−y) − eik(x+y)] dk.

(2.70)

The inverse Fourier transform may be expressed in terms of free particle solutions.This gives the following result.

Proposition 2.11. The propagator solution of the Schrodinger equation for aparticle confined to the interval [0,+∞] is given by

(2.71) pt(x, y) = giσ2t(x− y) − giσ2t(x+ y)

Thus for x ≥ 0 the solution is given by

(2.72) u(x, t) =

∫ ∞

0

pt(x, y)ψ(y) dy = (Utψ−)(x),

where Ut is the evolution operator for free motion and ψ− is the reflected initialcondition ψ−(y) = sign(y)ψ(|y|).

18 WILLIAM G. FARIS

This solution says that the solution is obtained by placing a mirror at the originand having the solution with initial condition at y matched by a solution with initialcondition at the reflected point −y that has the opposite sign. If a solution comes intraveling to the left, then it is eventually replaced by the reflected solution travelingright. The effect is that the solution coming in traveling to the left appears to bescattered at the origin, with a sign change, and to emerge traveling right.

2.9. The diffusion with drift representation. We have already seen thatthe Schrodinger equation is intimately related to a diffusion with removal equation.In this subsection we show that there is yet another representation as a diffusionwith drift equation. This is an important topic in probability theory, and it is strik-ing that it is so closely connected with quantum mechanics. See the introductorychapter in [5] or the book [11] for more information on this topic.

Let v(x) represent a continuous real function that is bounded below. Let

(2.73) −A =1

2σ2 ∂

2

∂x2− 1

~v(x).

As we have seen, the equation

(2.74)∂u

∂t= −Au

represents diffusion with removal at rate 1~v(x) when the diffusing particle is at

location x. It is shown in probability theory that this diffusion has an underlyingstochastic process, a probability measure on paths that describes diffusing particlesthat move in irregular paths, and sometimes vanish.

In the following, we shall often use an exponential notation for the solution ofsuch an equation, writing

(2.75) u(t) = e−tAψ.

This notation will later be justified by the spectral theorem for self-adjoint opera-tors.

Suppose that A has a smallest eigenvalue λ corresponding to an eigenvector φin the Hilbert space. Furthermore, suppose that φ(x) > 0 at each x. Then we canmake the change of variable

(2.76) L = φ−1(A− λI)φ.

Then

(2.77) −L =1

2σ2 ∂

2

∂x2+ σ2 φ

′(x)

φ(x)

∂

∂x.

Alternatively, we can make the change of variable

(2.78) L∗ = φ(A − λI)φ−1.

Then

(2.79) −L∗ =1

2σ2 ∂

2

∂y2− σ2 ∂

∂y

(

φ′(y)

φ(y)·)

.

These operators have an interpretation as diffusion with drift σ2φ′(y)/φ(y). Inaddition to the random diffusive motion there is a systematic drift depending onposition. The drift has units of velocity. It is shown in probability theory thatthis diffusion has an underlying stochastic process, a probability measure on paths


that describes diffusing particles that move in paths that are irregular but have atendency to drift in a direction given by the drift coefficient.

Why are there two such operators? Think of starting the diffusion at x andletting it evolve for time t > 0. The resulting probability density after the diffusionhas taken place is a function of y. The forward equation (Fokker-Planck equation)is

(2.80)∂

∂tw(y, t) = −L∗w(y, t).

It has solution

(2.81) w(y, t) = (e−tL∗

h)(y)

which describes the density as a function of the final y when the diffusion is startedat the random position h(x). The diffusion and drift combine to produce an equi-librium with probability density ρ(y) = φ(y)2. In fact, it is easy to check that

L∗ρ = 0 and e−tL∗

ρ = ρ.The backward equation is

(2.82)∂

∂tu(x, t) = −Lu(x, t).

It has solution

(2.83) u(x, t) = (e−tLf)(x)

which describes the describes the expectation of a function f of the final position asa function of the initial point x. One can even combine the forward and backwardsolutions to get the expectation of a function f of the final position when the initialpoint is random with density h. This is

(2.84) 〈h, e−tLf〉 = 〈e−tL∗

h, f〉.As an example, consider the particle in the box [0, a]. The smallest eigenvalue

of the generator A is λ1 = 12σ

2π2/a2 with eigenfunction φ1(x) = sin(πx/a). So the

drift term is σ2 πa

cot(πx/a). It is very positive near 0 and very negative near a.So the particle diffuses, but it is repelled from the end points, leading to diffusiveequilibrium with the largest part of the probability near the center of the interval.

In conclusion, there is a deep mathematical connection between quantum me-chanics and diffusion. The solution of the Schrodinger equation is e−itA. Thesolution of the diffusion with removal equation is given by e−tA. The solutions ofthe forward and backward equations for diffusion with drift are e−tL

∗

and e−tL.Moreover, since A − λI and L and L∗ are similar, they have the same spectralproperties.

2.10. The harmonic oscillator. One of the most famous quantum mechani-cal systems is the harmonic oscillator. This has many special properties that do notgeneralize to more complicated systems. However it is a landmark in the subject.The potential energy for the harmonic oscillator is the quadratic expression

(2.85) v(x) =1

2mω2x2.

The operator that occurs in the Schrodinger equation is then

(2.86) A = −1

2σ2 ∂

2

∂x2+

1

2

ω2

σ2x2.

20 WILLIAM G. FARIS

This subsection will be devoted to establishing formulas for the solution operatore−itA. As usual, we may express the solution operator in terms of a propagatorpt(x, y). In this case the expression for the propagator is complicated, but a briefinspection of the formula shows that it is indeed a function that is periodic in twith angular frequency ω.

Proposition 2.12. The propagator solution to the quantum harmonic oscilla-tor equation is given by the Mehler formula

(2.87) pt(x, y) =

√

ω

2πiσ2 sin(ωt)exp(i

ω

2σ2 sin(ωt)[cos(ωt)(x2 + y2) − 2xy]).

The spectral formula is also somewhat complicated, but again it exhibits thefact that solutions are periodic in t with angular frequency ω.

Proposition 2.13. Let g(x) denote the Gaussian function g σ2

2ω

(x). Let φ(x) >

0 be such that φ(x)2 = g(x). Let χn(x) = φ(x)−1(− ∂∂x

)nφ(x)2. Then the spectralsolution to the harmonic oscillator equation is given by

(2.88) pt(x, y) =

∞∑

n=0

1

n!

(

σ2

2ω

)n

χn(x)e−i(n+ 1

2 )ωtχn(y).

The function φ(x) is the ground state wave function, and g(x) = φ(x)2 is thecorresponding ground state position probability density. The eigenfunctions χn(x)are expressed in terms of the ground state wave function φ(x). These eigenfunctionsdefine an isomorphism from L2(R) to a weighed ℓ2 space. The solution operator

e−itA is given by the isomorphism followed by multiplication by e−i(n+ 12ω)t followed

by the inverse isomorphism. It follows that the eigenvalues of A are λn = (n+ 12 )ω,

and the corresponding energy values are En = ~λn = (n+ 12 )~ω.

The remainder of this subsection contains the derivation of these formulas.They result from a single general formula for the solution. This formula says thatafter several changes of variables the solution of the quantum harmonic oscillatoris given by a rotation in the plane. Thus the dynamics of the quantum harmonicoscillator reduces to the dynamics of a classical harmonic oscillator. Indeed, thesolution for a classical harmonic oscillator has a very simple representation in phasespace (position and momentum together); it is just a rotation. If the classical phasespace for a single particle is represented by the complex z plane, then a rotationwith angular frequency ω is a map that sends z into e−iωtz. This kind of rotationis precisely what underlies the dynamics of the quantum harmonic oscillator.

Even though the dynamics is essentially classical, there is still something es-sentially quantum mechanical about this system. For example, in the stationarystate the position is random, in fact Gaussian distributed with variance σ2/(2ω).

The change of variables operator W is given by a multiplication operator fol-lowed by the Fourier transform followed by another multiplication operator. Letg(x) denote the Gaussian function g σ2

2ω

(x). Let the ground state wave function

φ(x) > 0 be such that φ(x)2 = g(x). Then the image of L2(R, dx) under multipli-cation by φ(x) is a space of rapidly decreasing functions. The Fourier transformsof these functions are entire analytic functions of a Fourier transform variable z.The operator W is defined by

(2.89) Wψ = g−1F φψ.


For each complex number w define a corresponding rescaling operator by

(2.90) (Rwf)(z) = f(wz).

This rescaling operator sends entire analytic functions to entire analytic functions.If w = e−iωt, then the corresponding rescalings rotate the entire functions aboutthe origin at angular frequency ω.

Proposition 2.14. The operator e−itA that gives the solution of the quantumharmonic oscillator equation is given in terms of rotation operators Re−iωt by

(2.91) We−itAψ = e−i12ωtRe−iωtWψ.

There is of course a corresponding result for the operators e−tA that give thesolution of the diffusion with removal equation. In this case the scalings are realscalings of the form Re−ωt . The proof goes through for this case; at the end we canjust replace t by it.

There are three change of variable operators that go into W . The first ismultiplication by φ. First note that φ > 0 is an eigenfunction of A with eigenvalueλ = 1

2ω. From this we can compute the corresponding forward diffusion with lineardrift operator L∗. It is given by

(2.92) −L∗ =1

2σ2 ∂

2

∂x2+ ω

∂

∂xx.

It describes the diffusion of a probability density when the drift toward the originis −ωx. It satisfies φ(A − λI)ψ = L∗φψ and hence φe−tAψ = e−λte−tL

∗

φψ.The next change of variable is from x to the Fourier transform variable z. Write

h = φψ. Say that u(t) = e−tL∗

h is the solution of

(2.93)∂u

∂t= −L∗u.

Take the Fourier transform

(2.94) u(z, t) =

∫ ∞

−∞

e−izxu(x, t) dy.

The equation becomes

(2.95)∂u

∂t+ ωz

∂u

∂z= −1

2σ2z2u.

It is convenient to make a third change of variable w = g(z)−1u. The equationthen becomes a conservation law

(2.96)∂w

∂t+ ωz

∂w

∂z= 0.

The solution of this equation may be found by integrating along the curves dz/dt =ωz. These are the curves z = eωtz0. The solution is constant along such curves andhence is given by

(2.97) w(z, t) = f(z0) = f(e−ωtz).

This proves the result that the solution is given by scaling.Next we work backward to compute the propagator. We have

(2.98) u(z, t) = g(z)g(e−tωz)−1h(e−ωtz) = gσ2t(z)h(e−ωtz),

22 WILLIAM G. FARIS

where the variance parameter is

(2.99) σ2t =

σ2

2ω(1 − e−2ωt).

Scaling and multiplication in the Fourier transform representation correspond toscaling and convolution in the x representation. So(2.100)

(e−tL∗

h)(x) =

∫ ∞

−∞

gσ2t(x− y′)h(eωty′)eωt dy′ =

∫ ∞

−∞

gσ2t(x− e−ωty)h(y) dy.

From this we can compute e−tA. The result is the Mehler formula in the form(e−tAψ)(x) =

∫

kt(x, y)ψ(y) dy, where

(2.101) kt(x, y) =

√

ω

2πσ2 sinh(ωt)exp(− ω

2σ2 sinh(ωt)[cosh(ωt)(x2 + y2) − 2xy]).

As an immediate consequence we also have the corresponding Mehler formula forthe propagator for the quantum harmonic oscillator.

We can also begin to see the spectral properties. If we compute the actionof Re−ωt on zn we get e−nωtzn. From this we see that the eigenvalues are e−nωt.Higher powers decay more rapidly. For the quantum mechanical problem the actionof Re−iωt on zn gives e−inωtzn. In this case higher powers oscillate more rapidly.

Now we want to undo all these transformations. Back in the original Fouriertransform representation we write the eigenfunctions as (−iz)ng(z). (The (−i)nfactor is put in to make the final formulas take on a somewhat more familiar form.)

Back in x space this is(

− ∂∂x

)nφ(x)2. This eigenfunction of e−tL

∗

is a polynomialtimes a Gaussian. These polynomials are closely related to Hermite polynomials.Finally, we can go back to the original Hilbert space and the original e−tA. Theeigenvalues are e−(n+ 1

2 )ωt and the eigenfunctions are χn(x) = φ(x)−1(

− ∂∂x

)nφ(x)2.

Finally, in order to get the spectral representation of the propagator, it isconvenient to look at the operator e−tL

∗

φ2, the composition of the e−tL∗

withmultiplication by φ2 = g. The advantage of this operator is that it is self-adjoint.In the Fourier transform representation multiplication by φ2 = g corresponds toconvolution by g(z). An easy computation shows that in the Fourier transform

representation e−tL∗

φ2 is an integral operator with kernel

(2.102) r(z, w) = g(z)g(e−ωtz)−1g(e−ωtz − w) = g(z)eσ2

2ωe−ωtzw g(w).

Expand the exponential to get

(2.103) r(z, w) =

∞∑

n=0

1

n!

(

σ2

2ω

)n

(−iz)ng(z)e−nωt(iw)ng(w).

Take the inverse Fourier transform. This leads to the conclusion that e−tL∗

φ2 is anintegral operator with kernel

(2.104) r(x, y) =

∞∑

n=0

1

n!

(

σ2

2ω

)n

(− ∂

∂x)nφ(x)2e−nωt(− ∂

∂y)nφ(y)2.

From this it is immediate that e−tA is an integral operator with kernel

(2.105) kt(x, y) =

∞∑

n=0

1

n!

(

σ2

2ω

)n

χn(x)e−(n+ 1

2 )ωtχn(y).


This leads immediately to the spectral formula for the propagator for the quantumharmonic oscillator.

3. Self-adjoint operators

3.1. Stone’s theorem. This subsection features Stone’s theorem, which givesa correspondence between one-parameter unitary groups satisfying a continuitycondition and self-adjoint operators.

Consider a one-parameter group t 7→ Ut of unitary operators acting in theHilbert space H. Suppose also that it satisfies the continuity condition: for eachvector ψ in the Hilbert space, the map t 7→ Utψ is continuous from the real lineto the Hilbert space H. Consider the linear subspace D of H of all ψ such thatt 7→ Utψ is differentiable as a function from the real line to H. Then it may beshown that D is dense in H and that Ut sends D to itself. Furthermore, there isa unique linear operator A defined on the domain D with values H satisfying theabstract Schrodinger equation

(3.1) idUtψ

dt= AUtψ.

Stone’s theorem says that the operators A that arise in this way are preciselythe self-adjoint operators, and that furthermore the self-adjoint operatorA uniquelydetermines the unitary group Ut. In view of this result, it is natural to denote theunitary group determined by the self-adjoint operator A by Ut = e−itA. It willturn out that this notion of exponential is compatible with the notion given by thespectral theorem.

An operator A is self-adjoint if it is equal to its adjoint A∗. The following isa description of the general notion of adjoint operator. Let H be a Hilbert space.Let D(A) be a dense linear subspace of H. Let H′ be another Hilbert space, andlet A : D(A) → H′ be a linear operator. Then A is said to have adjoint operatorA∗ : D(A∗) → H with domain D(A∗) ⊆ H′ if the following conditions are satisfied.

(1) For every φ in D(A∗) and ψ in D(A) we have 〈A∗φ, ψ〉 = 〈φ,Aψ〉.(2) If there are φ and χ such that for all ψ in D(A) we have 〈χ, ψ〉 = 〈φ,Aψ〉,

then φ is in D(A∗) and A∗φ = χ.

The notion of adjoint operator is fundamental in Hilbert space theory. Hereare some examples. In the first two examples we must have H′ = H. In the lasttwo examples the domain is the entire Hilbert space.

• A is self-adjoint if A∗ = A.• E is an orthogonal projection if E∗ = E and E2 = E.• U is unitary if U∗ = U−1.

For another interesting example consider a vector χ in H. It defines a lineartransformation z 7→ zχ from C to H. It is natural to denote the adjoint transfor-

mation by χ∗. By the definition of adjoint 〈χ∗φ, z〉 = 〈φ, zχ〉 = z〈φ, χ〉 = 〈χ, φ〉z.This proves that χ∗φ = 〈χ, φ〉. So χ∗ is a linear transformation from H to C.

The considerations are relevant to Stone’s theorem. Suppose that A is self-adjoint, so the adjoint of A is A. Then the adjoint of −iA is iA. This makes itplausible that the adjoint of e−itA is eitA, that is, the adjoint of Ut is U−t = U−1

t .In other word, each Ut is unitary. In fact, all of this is true, and the situation issummarized in the following theorem.

24 WILLIAM G. FARIS

Theorem 3.1 (Stone’s theorem). Suppose there is a one-parameter unitarygroup t 7→ Ut that satisfies the continuity condition. Then there exists a uniqueself-adjoint operator A that generates this group via Ut = e−itA. Conversely, eachself-adjoint operator A determines a unique unitary group Ut = e−itA satisfying thecontinuity condition.

This remarkable result is ample justification for the study of self-adjoint op-erators. In particular, it shows that any reasonable time dynamics for a quantumsystem must be given by a self-adjoint operator. See [9] for a proof.

3.2. Multiplication operators. Multiplication operators are simple and con-crete, but they play a crucial role in operator theory. Suppose there is a spaceL2(K,µ) consisting of functions that are square-integrable with respect to somemeasure. The measure might be continuous or discrete. In the continuous case wewould write the square of the norm as

(3.2) ‖f‖2 =

∫

|f(k)|2 µ(dk).

In the discrete case we would have

(3.3) ‖f‖2 =∑

|f(k)|2 µ({k}).

There is also the possibility of mixed situations involving part integral and partsum. The notion of integral encompasses all such cases.

Let α be a complex function on K. Then there is a corresponding operatormultα that sends the function f in L2(K,µ) to the pointwise product α · f . If α isa bounded function, then multα is defined on all functions in L2. In general, multαis defined on the domain consisting of all f in L2 such that α · f is also in L2.

Consider the complex conjugate function α. This also defines a multiplicationoperator multα with the same domain. These two operators are adjoint; in partic-ular 〈α · f, g〉 = 〈f, α · g〉. If α is a real function, then multα is self-adjoint. If αonly takes on the values 0 and 1, then it is an orthogonal projection. If |α| = 1,then multα is unitary.

We say that an operator A is isomorphic to another operator B if there is aHilbert space isomorphism U such that A = U−1BU . Note that in particular, Umust be a bijection between the domain of A and the domain of B.

Proposition 3.1. Every real multiplication operator is self-adjoint. As a con-sequence, every operator that is isomorphic to a real multiplication operator is self-adjoint.

Here are some examples in the case when the Hilbert space is L2(R, dx).

• The first order differential operator −i ∂∂x

= F−1multkF is self-adjoint. Its

role in quantum mechanics is that the operator −i~ ∂∂x

is isomorphic viathe Fourier transform to multiplication by p = ~k. This is the momentumoperator.

• The second order differential operator − ∂2

∂x2 = F−1multk2F is self-adjoint.This is the square of the operator in the first example. Its role in quantum

mechanics is that the operator − ~2

2m∂2

∂x2 is isomorphic via the Fourier

transform to multiplication by p2

2m . This is the kinetic energy operator.


• Let v(x) be a real function on the line. Then multv(x) is a self-adjointoperator. This may be thought of as a potential energy operator. Oftenit is denoted more briefly as v(x).

• The operator − ~2

2m∂2

∂x2 + 12mω

2x2 is an operator for the total energy ofthe quantum harmonic oscillator. We have seen that it isomorphic tomultiplication by (n+ 1

2 )~ω on a weighted ℓ2 space of sequences.

The last example is a special case of the Schrodinger operator

(3.4) − ~2

2m

∂2

∂x2+ v(x).

This is the total energy operator in quantum mechanics. In general it is a difficultproblem to know whether or not this operator is self-adjoint. One way to provethat such an operator is self-adjoint would be to find an explicit isomorphism witha multiplication operator. However, this is usually impossible to accomplish. Forinstance, if v(x) = λx4 with λ > 0, then the operator is self-adjoint, but it re-quires heroic effort to get even partial information about an isomorphism with amultiplication operator. In the case λ < 0, there is no unambiguous definition asself-adjoint operator, because such a definition would require a choice of boundaryconditions at x = ±∞.

3.3. The spectral theorem. The spectral theorem says that every self-adjointoperator A is isomorphic to an operator that is multiplication by a real function.Here we state and explain this result. See [9] for a succinct proof.

Theorem 3.2 (Spectral theorem). Suppose A is a self-adjoint operator actingin a Hilbert space H. Then there is a space L2(K,µ), an isomorphism U : H →L2(K,µ), and a real measurable function α such that

(3.5) A = U−1multαU.

More specifically, if ψ is in H, then ψ is in the domain D(A) of A if and onlyif α · Uψ is in L2(K,µ), and in this case

(3.6) (UAψ)(k) = α(k)(Uψ)(k).

An isomorphism that makes A into a multiplication operator is called a spectralrepresentation. A spectral representation is not unique. One way to get a newone is to make a change of labeling. For example, suppose that g : K ′ → K is ameasure space isomorphism that carries the measure µ′ to the measure µ. Define anisomorphism W : L2(K,µ) → L2(K ′, µ′) by (Wf)(k′) = f(g(k′)). Similarly, definea new multiplication operator by α′(k′) = α(g(k′)). Then WU : H → L2(K ′, µ′)defines a new spectral representation that makes A isomorphic to multiplication byα′.

Another way to get a new spectral representation is to change the measure viaa density. Say that ρ is a measurable function such that ρ(k) > 0 for all k. Definea new measure µ by

(3.7)

∫

h(k) µ(dk) =

∫

h(k)ρ(k)µ(dk).

Define a Hilbert space isomorphism R : L2(K,µ) → L2(K, µ) by (Rf)(k) =

ρ(k)−12 f(k). Then RU : H → L2(K, µ) defines a new spectral representation that

makes A isomorphic to multiplication by α.

26 WILLIAM G. FARIS

The power of the spectral theorem is that it permits taking functions of a self-adjoint operator. Let f be a complex function on the line. Then f(A) is defined asthe operator that has spectral representation given by multiplication by f(α). Itmay be proved that this definition of a function of A is independent of the spectralrepresentation. Here is an important special case. It is the part of Stone’s theoremthat says that a self-adjoint operator determines a unitary group.

Proposition 3.2. Let A be a self-adjoint operator. Then Ut = e−itA as definedby the spectral theorem is a one-parameter unitary group.

The proof is to note that Ut = e−itA is isomorphic to multiplication by e−itα,which is unitary for each real t. Furthermore, the continuity condition is satisfied:for each vector ψ the map t 7→ Utψ is continuous. This continuity condition isnot difficult to prove from the spectral representation by an application of thedominated convergence theorem.

Let A be a self-adjoint operator. Suppose that S is a subset of the line, and1S is the indicator function of this set. Then 1S(A) is a well-defined self-adjointoperator, called the spectral projection of A corresponding to the subset S. We saythat λ is in the spectrum of A if for every ǫ > 0 the projection 1(λ−ǫ,λ+ǫ)(A) is anon-zero projection. We say that λ is in the point spectrum of A if 1{λ}(A) is anon-zero projection. If λ is in the point spectrum of A, then λ is an eigenvalue ofA, and 1{λ}(A) projects onto the corresponding eigenspace. Finally, we say thatλ is in the continuous spectrum of A if λ is in the spectrum but not in the pointspectrum. Here are examples.

• The first order differential operator −i ∂∂x

= F−1multkF has the samespectrum as multk. This is continuous spectrum including all of R. Thesame is true for the momentum operator −i~ ∂

∂xisomorphic to multiplica-

tion by p = ~k.

• The second order differential operator − ∂2

∂x2 = F−1multk2F has the same

spectrum as mult2k. This is continuous spectrum constituting the semi-infinite interval [0,+∞). The same is true for the kinetic energy operator

− ~2

2m∂2

∂x2 isomorphic to multiplication by p2

2m .• Let v(x) be a real function on the line. Then multv(x) is a self-adjoint

operator. Its spectrum is the essential range of the function v. Thespectrum may be a combination of continuous and point spectrum. Thepoint spectrum would corresponding to portions of the graph of v(x) thatare flat. The values at such points are eigenvalues of infinite multiplicity.

• The harmonic oscillator total energy operator − ~2

2m∂2

∂x2 + 12mω

2x2 has thesame spectrum as multiplication by the diagonal operator with entries(n + 1

2 )~ω on a weighted ℓ2 space of sequences. This is point spectrum,eigenvalues of multiplicity one.

3.4. Spectral measures. Consider a self-adjoint operator A and a state vec-tor ψ (a unit vector in the Hilbert space). The spectral theorem provides a unitaryoperator U : H → L2(K,µ) and a real function α defined on a space K equippedwith a measure µ. Furthermore, the measure |(Uψ)(k)|2µ(dk) is a probability mea-sure on K. Thus α is a random variable in the standard sense of mathematicalprobability theory, that is, a real function defined on a probability space.

As usual, the random variable has a distribution ν, a measure on the real line.This is defined by requiring that for each bounded real function h on the line the


corresponding expectation is

(3.8)

∫

h(α(k))|(Uψ)(k)|2 µ(dk) =

∫ ∞

−∞

h(x) ν(dx).

In other words, it is the image of the probability measure |(Uψ)(k)|2 µ(dk) underα.

The distribution may also be called a spectral measure, since it is non-zeroonly on the essential range of α, which is the spectrum of A. Since h(A) is a well-defined self-adjoint operator, the expectation with respect to this distribution maybe written in Hilbert space language as(3.9)

Eψ [h(A)] = 〈ψ, h(A)ψ〉 =

∫

(Uψ)(k)h(α(k))(Uψ)(k)µ(dk) =

∫ ∞

−∞

h(x) ν(dx).

There is nothing to prevent the spectral measure ν from being discrete. Infact, when the self-adjoint operator has point spectrum eigenvalues correspondingto eigenvectors in the Hilbert space, then for every state the spectral measure isnecessarily concentrated on these eigenvalues. One of the early surprises of quantummechanics was that this can occur for a self-adjoint operator that represents energy.This happens, for instance, for the particle in the box or for the harmonic oscillator.These discrete energy levels are responsible for the term “quantum” in this subject.They are seen in experimental observations of the energies. More precisely, in ascattering experiment that probes such a system the only changes in the kineticenergy between the ingoing and outgoing probe particles would be by eigenvaluedifferences of this energy operator.

Say that ψ is in the operator domain D(A) of A. Then we can define theexpectation (mean) and second moment of A. Thus

(3.10) Eψ[A] = 〈ψ,Aψ〉 =

∫

α(k)|(Uψ)(k)|2 µ(dk) =

∫ ∞

−∞

x ν(dx).

and

(3.11) Eψ [A2] = ‖Aψ‖2 =

∫

α(k)2|(Uψ)(k)|2 µ(dk) =

∫ ∞

−∞

x2 ν(dx)..

Then as usual the variance is given by

(3.12) (∆A)2ψ = Eψ [(A− Eψ[A])2] = Eψ[A2] − Eψ [A]2.

The second moment is an upper bound for the variance. The standard deviation isdefined as the square root of the variance.

Define the form domain Q(A) = D(|A| 12 ). We have D(A) ⊆ Q(A), but ingeneral the form domain is larger. Then for ψ in the form domain of A we can atleast define the expectation (mean) by

(3.13) Eψ [A] = 〈ψ,Aψ〉 =

∫

α(k)|Uψ(k)|2 µ(dk) =

∫

x ν(dx).

Strictly speaking, the notation 〈ψ,Aψ〉 should be interpreted as 〈|A| 12ψ, sign(A)|A| 12ψ〉.In a similar way we can define the form 〈χ,Aψ〉 for χ and ψ each in Q(A). Again

〈χ,Aψ〉 means 〈|A| 12χ, sign(A)|A| 12ψ〉.

28 WILLIAM G. FARIS

3.5. Generalized vectors. Generalized vectors give another perspective onthe unitary operators that occur in spectral representations. Here is a brief account.Consider an isomorphism U : H → L2(K,µ). Choose a vector ψ in H. For eachpoint k in the set K there is an evaluation (Uψ)(k) that is defined for almost everyk in K. With some optimism one may hope that the exceptional set may be takenindependent of ψ, so that the map ψ 7→ (Uψ)(k) is defined for almost every k and islinear in ψ. In general this map will not be continuous in ψ. In this circumstance,it is convenient to write the linear functional in the form

(3.14) (Uψ)(k) = 〈φk, ψ〉.In the case when the map is not continuous the φk will not be given by vectors inthe Hilbert space H, and the inner product is not the usual inner product in thisspace.

Here is a framework that makes sense of this notation. The vectors ψ belongsto some smaller Hilbert space H+ ⊆ H that is dense in H, and the φk belong tosome larger Hilbert space H− with H ⊆ H− and in which H is dense. The pairingbetween H− and H+ given by the brackets is conjugate linear in the first variableand linear in the second variable, and it coincides with the usual inner product onpairs taken from H.

Here is an example that is common when dealing with Schrodinger operators.Let ρ > 0 be a bounded real function with finite integral over the line. Thesmaller Hilbert space is H+ = L2(R, ρ(x)−1 dx), the original Hilbert space is H =L2(R, dx), while the bigger Hilbert space is H− = L2(R, ρ(x) dx). If φ is in H− andψ is in H+, then the pairing that agrees with the usual inner product on H is 〈φ, ψ〉.Since this may also be written in terms of an H inner product as 〈ρ 1

2φ, ρ−12ψ〉, it

follows from the Schwarz inequality that it is finite.A classic example where this choice of Hilbert spaces works is for the Schrodinger

operator for free motion. Then the spectral representation is given by the Fouriertransform. The φk in this case is the function φk(x) = eikx. This is not in H, but itis in H−. Furthermore, the Fourier transform is determined by the condition that

(3.15) (Fψ)(k) = 〈φk, ψ〉 =

∫ ∞

−∞

e−ikxψ(x) dx

for all ψ in H+.One common use of this notation is to express quantities like

(3.16) 〈χ, ψ〉 =

∫

(Uχ)(k)(Uψ)(k)µ(dk).

or

(3.17) 〈χ,Aψ〉 =

∫

(Uχ)(k)α(k)(Uψ)(k)µ(dk)

or

(3.18) 〈χ, e−itAψ〉 =

∫

(Uχ)(k)e−itα(k)(Uψ)(k)µ(dk)

It is reasonable to write 〈χ, φk〉 for the complex conjugate of 〈φk, χ〉. Then theequations become

(3.19) 〈χ, ψ〉 =

∫

〈χ, φk〉〈φk, ψ〉µ(dk).


or

(3.20) 〈χ,Aψ〉 =

∫

〈χ, φk〉αk〈φk, ψ〉µ(dk)

or

(3.21) 〈χ, eitAψ〉 =

∫

〈χ, φk〉e−itαk〈φk, ψ〉µ(dk)

It is also reasonable to introduce the notation φ∗k for the possibly discontinuouslinear map ψ 7→ 〈φk, ψ〉. One can then identify φk with the complex conjugate of

this map, so χ 7→ 〈χ, φk〉 = 〈φk, χ〉 is conjugate linear. The identity

(3.22) 〈χ,U−1f〉 =

∫

〈χ, φk〉f(k)µ(dk)

suggests writing

(3.23) U−1f =

∫

φkf(k)µ(dk).

If one is willing to interpret the expressions in a suitable weak form, then onecan write the spectral theorem is the particularly succinct form

(3.24) A =

∫

φkαkφ∗k µ(dk).

The propagator also has a expression

(3.25) e−itA =

∫

φke−itαkφ∗k µ(dk).

With the appropriate interpretation, this last formula is true in complete generality,and furthermore it is simple enough to be memorable. However, it should beremembered that it is just a summary of a statement about composition of unitarymaps.

The generalized vectors φk are mathematical objects that one can study in theirown right. They belong to a larger Hilbert space H− that includes H? How smallcan this space be and still be large enough to include these generalized vectors? See[2] for a possible answer. (It is sufficient that the injections from H+ to H and fromH to H− be Hilbert-Schmidt operators, but in specific cases it is often possible tomake do with a weaker condition.)

In theoretical work it may not be necessary to make explicit use of generalizedvectors, and it may even be clumsy to do so. All that is needed for quantum theoryis the isomorphism U from H to the Hilbert space to L2 and its inverse U−1. Thegeneralized vectors are hidden within the definition of each particular U , and theyoften need not be isolated as individual objects.

3.6. Dirac notation. The Dirac notation is a clever way of denoting vectorsand generalized vectors. Consider an isomorphism U : H → L2(K,µ). One canthink of the set K as a set of labels. Implicitly, one assumes that different spectralrepresentations involve different label sets. In the Dirac notation one writes

(3.26) (Uψ)(k) = 〈k | ψ〉.It is natural to define 〈χ | k〉 = 〈k | χ〉. Then in the Dirac notation

(3.27) 〈χ, ψ〉 =

∫

〈χ | k〉〈k | ψ〉µ(dk).

30 WILLIAM G. FARIS

and

(3.28) 〈χ,Aψ〉 =

∫

〈χ | k〉αk〈k | ψ〉µ(dk).

Sometimes one encounters expressions like 〈k | in isolation. It is clear whatthis means: it is a possibly discontinuous linear mapping from vectors ψ to scalarsgiven by 〈k | ψ〉 = (Uψ)(k). Similarly, | k〉 is the corresponding conjugate linearmap given by complex conjugation of the linear map. In this spirit, sometimes onesees expressions like

(3.29) I =

∫

| k〉〈k | µ(dk).

and

(3.30) A =

∫

| k〉αk〈k | µ(dk).

One can also write

(3.31) U−1f =

∫

| k〉f(k)µ(dk).

Consider another spectral representation given by (V ψ)(ℓ) = 〈ℓ | ψ〉. The compo-sition V U−1 is given by

(3.32) (V U−1f)(ℓ) =

∫

〈ℓ | k〉f(k)µ(dk).

The expression 〈ℓ | k〉 is called a bracket. In general it is not a function, but somekind of generalized function. The standard terminology in physics is that 〈ℓ | iscalled a bra, and | k〉 is called a ket. Sometimes a ket is thought of as a generalizedvector, as suggested by the expression for U−1f above, which writes a vector as asuperposition of kets.

Here are some examples. In the Dirac notation the inverse Fourier transformmight be written

(3.33) 〈x | ψ〉 =

∫ ∞

−∞

〈x | k〉〈k | ψ〉dk2π.

Here ψ is a vector in some abstract Hilbert space, 〈x | ψ〉 is its representation as afunction of position x, and 〈k | ψ〉 is its representation as a function of wave numberk. The bracket 〈x | k〉 is just another notation for the eikx factor that gives thetransformation between wave number and position representations. It is possibleto distinguish x values from k values because they have different units.

Another common situation is when a vector ψ is represented by coefficients interms of an orthonormal basis φn, where n runs over some index set. The coefficientsare 〈n | ψ〉 = 〈φn, ψ〉. Since 〈ψ | n〉 = 〈ψ, φn〉, it is natural to identify the ket | n〉with the unit vector φn. So the expansion of the vector is

(3.34) ψ =∑

n

| n〉〈n | ψ〉.

If we wanted this in the position representation, it would become

(3.35) 〈x | ψ〉 =∑

n

〈x | n〉〈n | ψ〉.

While in an example like the last one the ket | n〉 is a normalized basis vector,this is not the only possible situation. In fact, the wave number ket | k〉 is given


in the position representation by the function x 7→ eikx, which is not even in theHilbert space L2. This corresponds to the fact that the map ψ 7→ 〈k | ψ〉 givenby the corresponding bra is not continuous as a function on the Hilbert space H.Ultimately, this is because the value of the Fourier transform of a function at apoint is not continuous as a function on L2.

The clever feature of the Dirac notation is the use of the label set for therepresentation to denote various kinds of vectors and generalized vectors. However,the only place in science where the Dirac notation is in common use is in quantumtheory, and it is not necessary even there.

3.7. Spectral representation for multiplicity one. The spectral theoremas stated above is somewhat mysterious, since there are many possible spectral rep-resentations. It would be nice to find a standard form for a spectral representation.This would also shed light on how one could prove the spectral theorem: constructthe standard representation explicitly.

Let A be a self-adjoint operator and let Ut = e−itA be the correspondingunitary group. We say that the unit vector ψ is a cyclic vector if the smallestclosed subspace that contains all Utψ is the whole Hilbert space H.

Not every self-adjoint operator has a cyclic vector. However, there is the fol-lowing decomposition into cyclic subspaces.

Proposition 3.3. Let A be a self-adjoint operator acting in H. Then H isan orthogonal direct sum of closed subspaces Hj that are each invariant under theunitary operators e−itA. Furthermore, the subspaces may be chosen so that thereare vectors ψj in Hj that are cyclic for these restrictions.

The proof of this proposition is easy. Choose an arbitrary unit vector ψ1 andconsider the smallest closed subspace H1 with ψ1 in it and invariant under thee−itA. The orthogonal complement of H1 is also invariant under the e−itA. Ifthis orthogonal complement is not the zero subspace, choose ψ2 in the orthogonalcomplement. Let H2 be the smallest closed subspace of the orthogonal complementwith ψ2 in it and invariant under the e−itA. Continue in this way until the result isa maximal family of non-zero cyclic subspaces. These will automatically span theentire Hilbert space.

From now on we focus on finding a standard form associated with the self-adjoint operator A and the cyclic vector ψ. Actually, we shall describe two standardforms. Here is the first.

Proposition 3.4. Consider a self-adjoint operator with a unit cyclic vectorψ. Then there is a spectral representation W : H → L2(R, ν). Here ν is theprobability measure that is the distribution associated with ψ and A. The spectralrepresentation is determined by the requirement that (We−itAψ)(x) = e−itx. Theoperator that represents A is multiplication by x.

There is a characterization of measures associated with the real line. Here is abrief review. As always, we consider subsets and functions to be measurable (Borelmeasurable) by default. We consider measures defined on subsets of the line thatare finite on bounded intervals. Thus each subset B ⊆ R has a measure ν(B) ≥ 0.The assignment of measures satisfies the usual requirements from measure theory.

Every such measure is given by a function F that is increasing and right con-tinuous. To say that F is increasing is to say that x ≤ y implies F (x) ≤ F (y).

32 WILLIAM G. FARIS

(In some accounts this is called non-decreasing.) To say that F is right-continuousis to say that the right hand limit limx↓a F (x) = F (a). We can write this moresuccinctly as F (x+) = F (x). The measure of a subset (a, b] ⊆ R is given by thedifference ν((a, b]) = F (b)− F (a). The function F associated with ν is determinedup to an arbitrary constant. Once one has specified the measure on such spe-cial subsets, then it is determined for all subsets. In particular, for a single pointν({b}) = F (b)−F (b−). Once the measure is available, there is also a correspondingintegral defined for bounded functions g by

(3.36)

∫ ∞

−∞

g(x) ν(dx) =

∫ ∞

−∞

g(x) dF (x).

In the special case when the measure of the whole real line ν(R) = 1, themeasure is a probability measure, the corresponding integral is the expectation,and there is a choice of the arbitrary constant so that the function F takes valuesin the unit interval. For the moment continue to consider the general case.

The function F may have both flat places and jumps, but nevertheless it has akind of inverse. Let I be the open interval from inf F to supF . For y in I define

(3.37) G(y) = sup{x | F (x) < y}.Then G is increasing and left continuous. Furthermore, a < G(y) ≤ b is equivalentto F (a) < y ≤ F (b).

This allows a remarkable representation of the measure ν in terms of Lebesguemeasure λ on the interval I. We have(3.38)ν((a, b]) = F (b) − F (a) = λ({y | F (a) < y ≤ F (b)}) = λ({y | a < G(y) ≤ b}).

It then follows that

(3.39) ν(B) = λ({y | G(y) ∈ B}).In addition, it may be shown that

(3.40)

∫ ∞

−∞

h(x) ν(dx) =

∫

I

h(G(y)) dy.

In other words, every Lebesgue-Stieltjes measure on the real line is the image ofLebesgue measure on an interval by some increasing function G.

This representation is not a perfect correspondence between measures, sincea point measure at a discontinuity of F corresponds to Lebesgue measure on asubinterval of I. (The mass of the point measure is the length of the subinterval.)However consider the case when F is continuous. Then this is not an issue. Thereis also no problem with places where F is flat, since these are places where themeasure is zero, and these correspond to single points in I, which have Lebesguemeasure zero. So in this case the two measure spaces (the real line R with ν andthe interval I with Lebesgue measure ν) are in quite natural correspondence.

Proposition 3.5. Consider a self-adjoint operator A with a unit cyclic vectorψ. Suppose A only has continuous spectrum. Then there is a spectral representationU : H → L2(I, λ). Here I is the unit interval, and λ is Lebesgue measure. Let F bethe increasing right-continuous function associated with the spectral measure. Let Gbe the inverse function (taken in the above sense) of F . The spectral representationis determined by the requirement that (Ue−itAψ)(y) = e−itG(y). The operator thatrepresents A is multiplication by G(y).


The meaning of this result is the following. If we consider self-adjoint operatorswith a cyclic vector (which implies spectral multiplicity one), then we have only twokinds of spectral representation. We have point spectrum with distinct eigenvalues;the spectral representation is given by eigenvectors in the Hilbert space. Or we havecontinuous spectrum; the spectral representation is given by Lebesgue measure onan interval (which may be taken to be the unit interval). In the continuous spec-trum case, the operator itself is represented by multiplication by an increasing realfunction γ. The spectrum of the operator is the essential range of γ. The distribu-tion in a state is a probability measure on the spectrum. It is a continuous measurethat may be rather complicated, depending on how complicated the function γ is.

3.8. Form sums and Schrodinger operators. Stone’s theorem implies thata self-adjoint Schrodinger operator generates a quantum dynamics, but how doesone show that a Schrodinger operator is self-adjoint? One method might be to writethe Schrodinger total energy operator as the sum of a kinetic energy operator anda potential energy operator. Each of these has an explicit spectral representationthat exhibits it as a self-adjoint operator. So all that is required is a mechanismof showing that the sum of two self-adjoint operators is self-adjoint. This is nottrue in general for unbounded self-adjoint operators, but when the operators areboth positive there is no problem. This observation gives a very general way ofdetermining a quantum dynamics.

There is another way to think of this issue. How could the sum of kinetic andpotential energies fail to determine the dynamics? The kinetic energy (which ispositive) could rush to plus infinity while the potential energy could rush to minusinfinity, and after a finite time there could be some kind of catastrophe. However,if the potential energy is positive, or even bounded below, then this cannot happen.The following discussion presents the details of this argument.

Suppose that A is a self-adjoint operator. Then it is a linear transformationfrom the dense linear subspace D(A) ⊆ H to H. Consider D(A) with a new Hilbertspace inner product given by

(3.41) 〈φ, χ〉(2) = 〈Aφ,Aχ〉 + 〈φ, χ〉.The norm is given by

(3.42) ‖φ‖2(2) = ‖Aψ‖2 + ‖ψ‖2.

This Hilbert space has a very different character from the original Hilbert space.From the point of view of the spectral theorem, the norm in this Hilbert space isrelated to the second moment of the random variable.

As an example, consider the Hilbert space L2(R, dx) with the usual inner prod-

uct. Let A = − ∂2

∂x2 . In the Fourier transform representation this becomes multi-

plication by k2. So the (2)-norm in this case is given by

(3.43) ‖ψ‖2(2) = ‖∂

2ψ(x)

∂x2‖2 + ‖ψ‖2 = ‖k2ψ(k)‖2 + ‖ψ(k)‖2.

Suppose that A is a positive self-adjoint operator. Then it defines a positivequadratic form on the form domain Q(A) = D(A

12 ). This includes the operator

domain D(A), but for an unbounded operator it is strictly larger. Consider the

corresponding bilinear form 〈A 12χ,A

12ψ〉, defined for χ and ψ in Q(A). Henceforth

this will be written in the more attractive form 〈χ,Aψ〉 . In the same spirit, thecorresponding quadratic form satisfies 0 ≤ 〈ψ,Aψ〉 <∞ for ψ in Q(A).

34 WILLIAM G. FARIS

Consider Q(A) with the new Hilbert space inner product

(3.44) 〈φ, χ〉(1) = 〈φ,Aχ〉 + 〈φ, χ〉.The norm is given by

(3.45) ‖φ‖2(1) = ‖A 1

2ψ‖2 + ‖ψ‖2.

From the point of view of the spectral theorem, the norm in this Hilbert space isrelated to the first moment of the corresponding random variable.

As an example, consider the Hilbert space L2(R, dx) with the usual inner prod-

uct. Let A = − ∂2

∂x2 . In the Fourier transform representation this becomes multi-

plication by k2. So the (1)-norm in this case is given by

(3.46) ‖ψ‖2(1) = ‖∂ψ(x)

∂x‖2 + ‖ψ‖2 = ‖kψ(k)‖2 + ‖ψ(k)‖2.

Theorem 3.3 (Form representation theorem). Let H be a Hilbert space. Leta(φ, χ) be a positive form defined on a dense linear subspace H(1) ⊆ H. Supposethat with the inner product

(3.47) 〈φ, χ〉(1) = a(φ, χ) + 〈φ, χ〉the space H(1) is a Hilbert space. Then there is a unique positive self-adjoint oper-ator A such that H(1) = Q(A) and a(φ, χ) = 〈φ,Aχ〉.

The form representation theorem gives a particularly simple way of constructingself-adjoint operators. Say that A and B are self-adjoint operators. In general it isnot clear that the sum A+B makes sense. However, if A and B are both positive,then in very general circumstances one can use the form representation theorem todefine A+B as a self-adjoint operator.

Theorem 3.4 (Positive form sum theorem). Suppose that A and B are positiveself-adjoint operators. Suppose that Q(A) ∩ Q(B) is dense in H. Then there is aunique positive self-adjoint operator A + B whose form is the sum of the forms ofA and B.

This form sum result has a simple generalization to self-adjoint operators thatare bounded below. One simply has to add appropriate constants to reduce thiscase to the case of positive self-adjoint operators.

Here is an application to quantum mechanics. Consider the Hilbert spaceH = L2(R, dx) with the usual Lebesgue measure. Let P be the momentum operator.This is isomorphic via the Fourier transform to multiplication by p = ~k. So P hasthe simple explicit form

(3.48) P = −i~ ∂

∂x.

The kinetic energy operator is

(3.49) H0 =1

2mP 2.

Then H0 is isomorphic in the Fourier transform representation to multiplication by1

2mp2 = ~

2

2mk2. It has the explicit form

(3.50) H0 = − ~2

2m

∂2

∂x2.


The form of H0 for ψ in Q(H0) = D(P ) is

(3.51) 〈χ,H0ψ〉 =1

2m〈Pχ, Pψ〉 =

~2

2m

∫

∂

∂xχ(x)

∂

∂xψ(x) dx.

The derivative may be defined by the Fourier transform. This implies that ψ(x) isan absolutely continuous function, that is, an indefinite integral of another function,and that both ψ and ∂ψ/∂x are in L2(R).

Let the position operator Q be multiplication by x. If v is a real function onthe line, define the potential energy operator V to be the operator v(Q). This ismultiplication by v(x), and so it is also a self-adjoint operator. The form of V isdefined for ψ in Q(V ) by

(3.52) 〈χ, V ψ〉 =

∫

χ(x)v(x)ψ(x) dx.

Suppose that v is positive (or more generally, bounded below). The total energyis

(3.53) H = H0 + V =P 2

2m+ v(Q) = − ~

2

2m

∂2

∂x2+ v(x).

To use the method of form sums find a condition that guarantees that Q(H0)∩Q(V )is dense in L2(R). For instance, take the dense set to be smooth functions withcompact support. These are clearly in Q(H0). If v is locally integrable, then theyare also in Q(V ). This proves the following result.

Proposition 3.6. Suppose that v(x) is bounded below and locally integrable.

Then the Schrodinger operator H = P 2

2m + v(Q), defined as a form sum, is a well-defined self-adjoint operator.

One consequence of this result is that the unitary operator e−itH

~ is a well-defined unitary evolution that gives a solution of the Schrodinger equation. Formore on form sums see the treatise [8]. The book [10] give many applications toquantum mechanics. For a brief account see [3].

4. The role of Planck’s constant

4.1. The uncertainty principle. The Heisenberg uncertainty principle isperhaps the most famous assertion of quantum mechanics. In this subsection weshall see that this principle, interpreted suitably, has important physical interpre-tations. In particular, it shows that in some circumstances negative singularities ofthe potential energy are completely harmless.

There is particularly simple and symmetric mathematical formulation of theuncertainty principle that makes it an elementary result of Fourier analysis. Thisis not a strong enough version of the principle to give the desired physical result,but it is a place to start.

Proposition 4.1 (Heisenberg uncertainty principle). For every unit vector ψthe standard deviations of position and wave number satisfy the inequality

(4.1) (∆x)ψ(∆k)ψ≥ 1

2.

Since in quantum mechanics the position Q is multiplication by x and the momen-tum P is given in the Fourier transform representation by multiplication by p = ~k,

36 WILLIAM G. FARIS

the principle also takes the form

(4.2) (∆Q)ψ(∆P )ψ ≥ ~

2.

The Heisenberg uncertain principle is a standard result. It is a consequence ofa simple identity together with the Cauchy-Schwarz inequality. However there isanother, closely related, uncertainty principle that is just as simple but much morepowerful in its physical applications.

Proposition 4.2 (Local uncertainty principle). For every unit vector ψ thecorresponding position probability density satisfies

(4.3) supx

|ψ(x)|2 ≤ (∆k)ψ.

In quantum mechanics this implies that the expectation of a function v(Q) of posi-tion satisfies

(4.4) ±Eψ[v(Q)] ≤(

∫ ∞

−∞

|v(x)| dx)

1

~(∆P )ψ.

The idea of this result is that concentration in momentum implies spread inposition. The spread in position is in a very strong sense: the position probabilitydensity near every given point is small. It is proved at the end of this subsection.

For every a > 0 there is an estimate

(4.5) (∆P )ψ ≤ Eψ[P 2]12 ≤ 1

2aEψ[P 2] +

a

2.

This shows that in every state the expected potential energy is bounded by

(4.6) ±〈ψ, v(Q)ψ〉 ≤(

∫ ∞

−∞

|v(x)| dx)

1

~

[

1

2a〈ψ, P 2ψ〉 +

a

2

]

.

An appropriate choice of a leads to the following lower bound.

Proposition 4.3. Let

(4.7) E =1

2

m

~2

(∫ ∞

−∞

|v(x)| dx)2

.

In every state ψ the total energy in quantum mechanics satisfies the lower bound

(4.8) 〈ψ, P2

2mψ〉 + 〈ψ, v(Q)ψ〉 ≥ −E.

The physical meaning of this application of the uncertainty principle is strik-ing. Suppose that the potential energy has a negative singularity, that is, it hasarbitrarily large negative values near some point. Then it may still be true thatthe total energy is bounded below. The reason is that the probability of being nearthe point cannot be too large, unless this is compensated by a large kinetic energy.

The same kind of reasoning gives a rigorous proof of self-adjointness for certainSchrodinger operators for which the potential energy is not bounded below. Thefollowing theorem is the key.

Theorem 4.1 (Relatively small form sum theorem). Suppose that A is a pos-itive self-adjoint operator. Suppose that B is a self-adjoint operator with Q(A) ⊆Q(B). Suppose that there exist positive constants ǫ < 1 and b such that for all ψ inQ(A) we have

(4.9) ±〈ψ,Bψ〉 ≤ ǫ〈ψ, (A+ b)ψ〉.


Then there is a unique self-adjoint operator A + B whose form sum is the sum ofthe forms of A and B.

This theorem is an easy consequence of the form representation theorem. Ithas an important consequence for Schrodinger operators. The hypothesis on thepotential energy function is that it is the sum of an integrable function and abounded function. In this case, one can take the integrable part arbitrarily smalland apply the uncertainty principle bound. The result is the following assertion.

Proposition 4.4. Suppose that v(x) is the sum of an integrable function and

a bounded function. Then the Schrodinger operator H = P 2

2m + v(Q), defined as aform sum, is a well-defined self-adjoint operator.

This subsection concludes with a proof of the local uncertainty principle. It isclear that

(4.10) |ψ(x)|2 ≤(

∫ ∞

−∞

|ψ(k)| dk2π

)2

.

By the Cauchy-Schwarz inequality

(4.11)

(∫ ∞

−∞

|ψ(k)| dk2π

)2

≤∫ ∞

−∞

1

k2 + a2

dk

2π

∫ ∞

−∞

(k2 + a2)|ψ(k)|2 dk2π.

These give

(4.12) |ψ(x)|2 ≤ 1

2a(‖kψ‖2 + a2) =

1

2a‖kψ‖2 +

a

2.

The case a = ‖kψ‖ gives a bound of the left hand side in terms of the square rootof the second moment. The bound in terms of the standard deviation follows froma shift in wave number, since such a shift does not change the absolute value of theposition wave function.

4.2. Classical mechanics. This subsection treats the case when Planck’sconstant may be neglected. This is often called the classical limit, but this ter-minology gives a misleading impression of the physical situation. Since Planck’sconstant ~ = 1.05443× 10−27erg s is a constant, the success of classical mechanicscannot be due to the fact that ~ approaches zero.

One way to think of the role of Planck’s constant is in terms of the originalform of the uncertainty principle due to Heisenberg. This is the statement that inany state, the product of standard deviations ∆p ∆q ≥ ~

2 . A state with localizedmomentum must have a large position standard deviation.

However there is nothing in general to prevent a state from having a reason-ably small position standard deviation. The idea is that a small position standarddeviation should be compatible with classical behavior. The question arises: smallcompared to what? To understand this issue, we begin with Ehrenfest’s theorem,which makes no reference at all to Planck’s constant.

Proposition 4.5 (Ehrenfest’s theorem). Let u(t) be a solution of the Schrodingerequation. Then

(4.13) md2〈u(t), qu(t)〉

dt2= −〈u(t), v′(q)u(t)〉.

38 WILLIAM G. FARIS

This theorem is true under appropriate hypotheses on the regularity of thesolution of the Schrodinger equation. It is proved by elementary computation. Theproblem with the result is that it does not give a closed form equation for theexpected position 〈u(t), qu(t)〉.

Let qt = 〈u(t), qu(t)〉 be the expected position. Expand v′(q) ≈ v′(qt) +v′′(qt)(q − qt) + 1

2v′′′(qt)(q − qt)

2. Inserting this in the equation, we get

(4.14) md2qtdt2

≈ −v′(qt) −1

2v′′′(qt)(∆q)

2t .

From this we see that a condition for neglecting the last term is that

(4.15) (∆q)t ≪ supx

√

|v′(x)||v′′′(x)| .

In other words, the condition is that the position standard deviation should besmall compared to a certain scale of variation of the potential energy function. Theconclusion is that the classical equation

(4.16) md2qtdt2

= −v′(qt)

applies, at least in some loose approximation. This argument is not rigorous, butit is illuminating, in that it hints at why classical behavior can be relevant, evenfor a fixed value of Planck’s constant.

Still, there is something odd about the situation. This argument shows at bestthat the expected position might come close to satisfying the classical law of motion.The expected position is only a statistical quantity. In orthodox quantum mechanicsthere is no attempt to characterize actual particle trajectories, even statistically.That is, there is no stochastic process underlying the motion of a quantum particle.So the quantum path cannot have a classical limit, because there is no quantumpath.

5. Spin and statistics

5.1. Spin 12 . In quantum mechanics angular momentum is a discrete vari-

able. Recall that Planck’s constant ~ has the dimensions of angular momentum. Itturns out that the values of angular momentum that can occur have magnitudes0, 1

2~, ~, 32~, 2~, 5

2~, . . .. Either sign can occur. Particles such as electrons, protons,

and neutrons have an intrinsic angular momentum of magnitude 12~. This is called

spin. The possible spin values in a given direction are ± 12~. So spin in a given

direction is like a variable representing a coin toss. In the following we will mostlytake the spin in units of 1

2~. Thus the values will be simply ±1.

The Hilbert space that describes the states of a spin 12~ particle is two-dimensional.

Write

(5.1) ψ =

(

ψ1

ψ2

)

.

for a vector in the Hilbert space C2. Then ‖ψ‖2 = |ψ1|2 + |ψ2|2. It is a unit vectorin C2 if this length is one. In this case it describes the quantum state. Such avector is called a spinor, since it has a geometrical significance. The space of allspinors forms a three-sphere in four-dimensional space. This three-sphere maps tothe two-sphere in three-dimensional space. In geometry this is called the Hopf map.


A spinor ψ in C2 that maps to a unit vector (x, y, z) in R3 determines a quantumstate that makes the spin in the direction (x, y, z) sure to have the value +1. It isthe ability to freely choose the phase of the spinor in determining the correspondingstate that allows the reduction from the 3-sphere down to the 2-sphere.

An easy way to define this map is to use spherical polar coordinates for three-dimensional space. We have z = r cos(θ), x = r sin(θ) cos(φ), and y = r sin(θ) sin(φ).Here θ is the colatitude and φ is the longitude. To get a unit vector in the two-sphere set r = 1. The spinor ψ describes a unit vector as follows. Write ψ in theform

(5.2) ψ =

(

ψ1

ψ2

)

=

(

cos(θ/2)eiχ1

sin(θ/2)eiχ2

)

.

Then θ is the colatitude, and the relative phase φ = χ2 − χ1 is the longitude.

Proposition 5.1. The map from the three-sphere of normalized spinors ψ inC2 to the two-sphere of unit vectors (x, y, z) in R3 is given by

(5.3) z = cos(θ) = cos2(θ/2) − sin2(θ/2) = |ψ1|2 − |ψ2|2

and

(5.4) x+ iy = sin(θ)eiφ = 2 cos(θ/2) sin(θ/2)e−iχ1eiχ2 = 2ψ1ψ2.

The inverse image of a unit vector in three-dimensional space is a circle. Thiscorresponds to an arbitrary phase in the spinor. This inverse image also has anexplicit representation.

Proposition 5.2. The unit vector (x, y, z) determines the corresponding spinorup to a phase by the formula

(5.5)

(

ψ1

ψ2

)

= eiχ11

√

1+z2

(

1+z2

x+iy2

)

= eiχ21

√

1−z2

(

x−iy2

1−z2

)

.

The relation between a vector R3 and a spinor is complicated by the fact thatthe vector is quadratic in the spinor. The next proposition will show that thereis a correspondence between the vector and the state defined by the spinor that islinear.

For this it is convenient to describe the spinors in terms of the Pauli matricesdefined by

(5.6) σx =

(

0 11 0

)

, σy =

(

0 −ii 0

)

, σz =

(

1 00 −1

)

.

These combine to make a vector ~σ whose components are these three matrices.Consider a vector ~u with components (x, y, z). Then the corresponding Pauli

matrix is

(5.7) ~σ · ~u =

(

z x− iyx+ iy −z

)

.

These matrices satisfy the algebraic relation

(5.8) (~σ · ~u)(~σ · ~v) = ~u · ~v I + i~σ · (~u × ~v).

In particular (~σ · ~u)2 = ~u · ~u I.The formula for the product of Pauli matrices should seem familiar to most

mathematicians. Recall that a quaternion is a sum t + ~u, where t is a scalar and

40 WILLIAM G. FARIS

~u is a vector in three dimensional space. It is obvious how to define the productof two quaternions, except for the term involving the product of two vectors. Thisquaternion product is given by combining the scalar dot product and the vectorcross product to give a quaternion

(5.9) ~u~v = −~u · ~v + ~u× ~v

If we represent the scalar t by t times the identity I, and if we represent the vector~u by −i~σ · ~u, then the resulting matrices form an representation of the quaternionalgebra.

Let ~u be a unit vector, so that (~σ · ~u)2 = I. Then ~σ · ~u represents the spinalong this axis. It has eigenvalues ±1. Let

(5.10) E(~u) =1

2(I + ~σ · ~u) =

1

2

(

1 + z x− iyx+ iy 1 − z

)

.

This E(~u) projects onto the eigenspace of ~σ · ~u with eigenvalue 1. The spinors inthis eigenspace map to the unit vector u. These represent states where the spincomponent in the u direction is 1. It is not hard to see that the spinors in theeigenspace with eigenvalue −1 map to the unit vector −u. These represent stateswhere the spin component in the u direction is −1.

Proposition 5.3. Each state vector ψ of the spin 1/2 system corresponding tothe unit vector ~u defines a corresponding projection operator ψψ∗. The correspon-dence between projection operators defining states and the unit vectors ~u is givenexplicitly by

(5.11) ψψ∗ = E(~u).

5.2. Composite systems. This subsection contrasts two constructions ofHilbert spaces. A direct sum of Hilbert spaces corresponds to disjoint union ofsets, while a tensor product of Hilbert spaces corresponds to cartesian productof sets. It is the tensor product construction that gives the quantum mechanicaldescription of composite systems.

Suppose that H is a Hilbert space and that H1 is a closed linear subspace ofH. Let H2 be the orthogonal complement of H1 in H. Then the projection theoremsays that H is the orthogonal direct sum H = H1

⊕

H2. Thus, for every vectorψ in H there is a vector φ in H1 and a vector χ in H2 with ψ = φ + χ, and thisdecomposition is unique. Finally, since φ is orthogonal to χ, we have the theoremof Pythagoras ‖ψ‖2 = ‖φ‖2 +‖χ‖2. Sometimes this is called an internal direct sum.

If instead we are given two Hilbert spaces H1 and H2, then we can form a newHilbert space H = H1

⊕

H2 that is an external direct sum. This consists of allordered pairs φ ⊕ χ with φ in H1 and χ in H2. The norm of a pair is defined bytaking the theorem of Pythagoras as a definition, so ‖φ⊕ χ‖2 = ‖φ‖2 + ‖χ‖2.

Say that the two Hilbert spaces are spaces of functions: H1 = L2(X1) andH2 = L2(X2). Then the direct sum H = L2(X1 ⊔ X1), where X1 ⊔ X2 is thedisjoint union. Thus if φ is a function on X1, and χ is a function on X2, then φ⊕χis the function on X1 ⊔X2 whose value on x is φ(x) if x ∈ X1 and χ(x) if x ∈ X2.

The tensor product construction is similar to the external direct sum. GivenHilbert spaces H1 and H2, there is a tensor product Hilbert space H = H1

⊗

H2.Furthermore, given vectors φ in H1 and χ in H2, there is always a vector φ⊗ χ inH2. However now the situation is more complicated; the vectors of this form neednot constitute the entire tensor product space. For instance, linear combinations


of such vectors also belong to the tensor product space. The inner product in thetensor product Hilbert space should satisfy 〈φ ⊗ χ, φ′ ⊗ χ′ = 〈φ, φ′〉〈χ, χ′〉. Againthis needs to be extended to linear combinations. All this can be done, but thedetails require some care.

One way to see what is going on is to represent the Hilbert spaces as H1 =L2(X1) and H2 = L2(X2). Then H = L2(X1×X2), where X1×X2 is the cartesianproduct. So a function in H is a function ψ(x1, x2) that is square-integrable withrespect to the product measure. If φ and χ are vectors in H1 and H2, then theirtensor product is

(5.12) (φ⊗ χ)(x1, x2) = φ(x1)χ(x2).

There is certainly no reason to expect that the general function of two variableswill have this product form. However, consider a vector ψ in the tensor productHilbert space. If it factors as ψ = φ⊗ χ, then it is called decomposable. Obviouslythis is a very special case.

In addition to tensor product vectors, there are tensor product operators. Forinstance, if A : H1 → H1 and B : H2 → H2, then there is an operator A ⊗ B :H → H determined by (A⊗B)(φ⊗ χ) = Aφ⊗Bχ. In the case when A and B areself-adjoint, it is also natural to consider the commuting operators A⊗ I and I⊗Band their sum A ⊗ I + I ⊗ B. Then there is a corresponding relation for unitaryoperators

(5.13) exp(−it(A⊗ I + I ⊗B)) = exp(−itA) ⊗ exp(−itB).

In quantum mechanics, the tensor product describes composite systems. Thus,for instance, if H1 = L2(X1) is the Hilbert space for particle 1, and H2 = L2(X2)is the Hilbert space for particle 2, then H = L2(X1 ×X2) is the Hilbert space forsystem consisting of the two particles. (Here the particles are assumed distinguish-able.) This leads to the remarkable conclusion that the wave function ψ(x1, x2)for the two particle system is a function of both particle position coordinates atonce. In general, the wave function ψ will not be decomposable. In that case, thetwo particle system is said to be entangled. There is no longer a well-defined wavefunction for the individual particles. This is the typical case.

Consider the decomposable case when ψ = φ⊗χ, that is, ψ(x1, x2) = φ(x1)χ(x2).Then it is possible to think of φ(x1) and χ(x2) as describing the states of the indi-vidual particles. However this case is very much the exception.

One famous case of the tensor product is the Hilbert space C2⊗

C2 describingtwo spins. There is an entangled state of the combined system called the singletstate. This is given by

(5.14) ψ0 =1√2

[ (

10

)

⊗(

01

)

−(

01

)

⊗(

10

) ]

.

The idea is that the spins are as opposite as they can get. We shall see in a latersection that this state is invariant under rotations.

Physicists often use the Dirac notation in the context of spin systems. Thusthey might use labels ↑ and ↓ for the vector indices. So the singlet state might bedenoted

(5.15) ψ0 =1√2

( |↑〉⊗ |↓〉− |↓〉⊗ |↑〉 ) .

42 WILLIAM G. FARIS

Or the indices for the tensor product states might be pairs of arrows, so the samestate would be

(5.16) ψ0 =1√2

( |↑↓〉− |↓↑〉 ) .

It is easy to compute the properties of spin in the singlet state. (See, forinstance, the appendix to [12].) Consider unit vectors ~u and ~v. There is anoperator ~σ · ~u ⊗ I for the ~u component of the first spin and operator I ⊗ ~σ · ~v forthe ~v component of the second spin. In the singlet state these each have values ±1with equal probabilities. Since they commute, they have a joint distribution. If ρis the angle between the unit vectors ~u and ~v, then the probability that they havethe same value is sin2(ρ/2), and the probability that they have the opposite valueis cos2(ρ/2). In particular, when ~u = ~v and ρ = 0 they are always opposite. Thissystem is a building block for many remarkable quantum effects exhibiting someform of non-locality.

People sometimes argue about whether quantum mechanics is non-local. Ofcourse, this depends on the meaning of the word “local.” (See the appendix to[12] for one analysis.) However there is already a rather evident sense in whichquantum mechanics is non-local, namely, that the wave function for two particlesis a single function of the two particle coordinates. This can have dramatic effectseven when the particles are widely separated in space, provided this wave functionis not decomposable. If the particles have spin, then their spin state can alsobe indecomposable, as in the example of the entangled singlet state. Compositesystems in quantum theory can behave very differently from what one would expectfrom their constituents.

5.3. Statistics. Another fact about quantum mechanics is the special treat-ment of composite systems of indistinguishable particles. There are two kinds ofsystems: Bose-Einstein (boson) and Fermi-Dirac (fermion). For Bose-Einstein sys-tems the state vectors of a composite system are restricted to be symmetric underinterchange, while for Fermi-Dirac systems they are restricted to be anti-symmetricunder interchange. Fermi-Dirac statistics are particularly important in atomic sys-tems, since they apply to electron systems. This leads to the famous Pauli exclusionprinciple, which says that two electrons cannot occupy the same quantum state.

Finally, there is a principle that connects spin and statistics. Identical parti-cles of spin for 0, 1, 2, . . . obey Bose-Einstein statistics, while identical particles ofspin 1

2 ,32 ,

52 , . . . obey Fermi-Dirac statistics. In particular, since electrons have spin

12 , they obey Fermi-Dirac statistics. Hence the Pauli exclusion principle for elec-trons. The connection between spin and statistics is both general and profound; seeFrohlich’s article “Spin and quantum statistics” [6, pp. 19–61] for a mathematicaldiscussion. Quantum mechanics is already an amazing subject; the extra featuresinvolving spin and statistics merely underline what a strange theory it is.

6. Fundamental structures of quantum mechanics

6.1. Self-adjoint operators as dynamics. The spectral theorem impliesthat for each self-adjoint operator A there is a corresponding one-parameter unitarygroup t 7→ Ut = e−itA. It satisfies the continuity condition: for each vector ψ in theHilbert space, the map t 7→ Utψ is continuous from the real line to the Hilbert space.Stone’s theorem also gives a converse statement: every one-parameter unitary group


satisfying the continuity condition arises in this way. If one believes that quantumdynamics is given by unitary operators, then this result shows why self-adjointoperators play such a central role in quantum theory.

A densely defined operator A : D(A) → H is said to be Hermitian if for eachφ and ψ in D(A) we have 〈Aφ,ψ〉 = 〈φ,Aψ〉. This is a purely algebraic property,but it is not enough to make the operator self-adjoint. For the operator to beself-adjoint there is an additional analytic condition. This says that if there are φand χ such that for all ψ in D(A) we have 〈χ, ψ〉 = 〈φ,Aψ〉, then φ is in D(A) andAφ = χ. The significance of this condition is explained in treatments of operatortheory, such as [3].

The adjoint operator of a Hermitian operator may have a larger domain, and ingeneral it will not be Hermitian. The usual interpretation of this situation is thatthe Hermitian operator A has too many boundary conditions, making its domaintoo small, while the adjoint A∗ has too few boundary conditions, making its domaintoo big. A self-adjoint operator has just the right boundary conditions needed toallow it to define an unambiguous dynamics.

It is possible that A is Hermitian (but not self-adjoint) andA∗ is also Hermitian.In that case A∗ is self-adjoint, and A is said to be essentially self-adjoint. Thiscorresponds to a case when the domain is restricted in some unimportant way, butstill defines the correct boundary conditions. When an operator is essentially self-adjoint, it determines a unique self-adjoint operator, and this self-adjoint defines aquantum dynamics as before.

The self-adjointness problem is to define the quantum dynamics e−itA for eachreal t. This is a problem of fixed time interval and large energy. Some of the mainresults were described in a previous section. Once this is accomplished, there is thetask of finding the asymptotic behavior of the dynamics for t → ±∞. This is thestudy of spectral theory and associated phenomena, such as scattering. This is aproblem of fixed energy interval and large time. This subject is too extensive tocover in an outline of limited scope.

6.2. Self-adjoint operators as states. This subsection treats very specialoperators: positive self-adjoint operators with trace one. These give a particularlyconvenient way of describing quantum states. First we need some preliminaryconsiderations.

An operator K is Hilbert-Schmidt if tr(K∗K) < ∞. This trace is defined inthe following way. Let φn be an orthonormal basis. Then

(6.1) trK∗K =∑

n

〈φn,K∗Kφn〉 =∑

n

‖Kφn‖2.

It may be shown that this is independent of the choice of orthonormal basis. Also,the adjoint K∗ is Hilbert-Schmidt, with trKK∗ = trK∗K. If K acts in an L2 space,then K is always given by an integral operator

(6.2) (Kf)(x) =

∫

k(x, y)f(y) dy.

The product W = K∗K is also an integral operator

(6.3) (Wf)(x) = (K∗K)f(x) =

∫ ∫

k(z, x)k(z, y)f(y) dz dy.

44 WILLIAM G. FARIS

It may be shown that in this case

(6.4) tr(K∗K) =

∫ ∫

|k(z, y)|2 dz dy.

So Hilbert-Schmidt operators are easy to characterize: they are integral operatorswith square-integrable kernels.

If W = M∗K is the product of two Hilbert-Schmidt operators M∗ and K,then it itself is a Hilbert-Schmidt operator. However not every Hilbert-Schmidtoperator arises in this way. An operator W is said to be trace class if it is theproduct W = M∗K of two Hilbert-Schmidt operators. Its trace is defined by

(6.5) trW = trM∗K =∑

n

〈φn,M∗Kφn〉 =∑

n

〈Mφn,Kφn〉.

If a trace class operator W acts on an L2 space, it is an integral operator

(6.6) W =

∫

w(x, y)f(y) dy

with square integrable kernel. One might hope that the trace would be defined bythe integral

(6.7) tr(W ) =

∫

w(x, x) dx.

On the other hand, it might seem too good to be true, since one is integratingover a set of measure zero in the product space. There is something delicate aboutchecking that an operator is trace class. However, ifW is represented asW = M∗K,and the kernel w(x, y) is correspondingly represented as

(6.8) w(x, y) =

∫

m(z, x)k(z, y) dz,

then the integral expression for the trace is valid.The class of operators of present interest are positive self-adjoint trace class

operators with trace equal to one. Thus W is such an operator if W ∗ = W andW ≥ 0 and trW = 1. By the spectral theorem these operators are of the form

(6.9) W =∑

n

λnχnχ∗n.

where the χn are unit vectors, and the λn ≥ 0 satisfy tr(W ) =∑

n λn = 1.Two unit vectors define the same quantum state if they differ by a phase.

So one can identify the state associated with the vector ψ with the orthogonalprojectionE = ψψ∗ onto the corresponding one-dimensional space. This orthogonalprojection is a very special kind of positive self-adjoint operator with trace one. Theexpectation of the self-adjoint operator A in this state is

(6.10) 〈ψ,Aψ〉 = tr(AE).

Consider a quantum mechanical system with a random state. This means thatthere is a given probability measure γ on the space of all states. The state spaceis identified as the space of all one-dimension orthogonal projections E. Form theoperator average

(6.11) W =

∫

Eγ(dE).


It may be shown that W is a positive self-adjoint operator with trace one. Theconventional term in quantum mechanics for such an operator is density matrix.The utility of this notion is due to the equation

(6.12)

∫

tr(AE)γ(dE) = tr(AW ).

The density matrix gives a convenient summary of the result of the averaging.One cannot go backward from the density matrix W to recover the measure

γ. It is true that the spectral representation of W produces a discrete measurewith masses λj on a set of states corresponding to orthogonal vectors χj. However,there is no guarantee that this is the measure that actually gave rise to the densitymatrix. There is no obstacle, for instance, to obtaining a random quantum systemby a random choice of state vectors that are not orthogonal.

Sometimes a density matrix is said to specify a mixed state. (In the casewhen it is a one-dimensional orthogonal projection it is a pure state. This is thenotion of state that we have been using up to now.) When a mixed state arisesby randomization, the actual randomization that is used is not determined by thestate.

There is another way that a density matrix can occur. Consider a subsystemthat is part of a larger system. The larger system may be in a pure state, butthis does not mean that the subsystem has a description via a pure state. Howeverobservations on the subsystem may always be described by a density matrix. In thiscase, it is called a reduced density matrix. The use of the reduced density matrixrepresents a loss of information about the original larger system.

As an example, take the case when we have two particles with entangled wavefunction ψ(x1, x2). Since this is a pure state, the density matrix is an integral

operator given by the integral kernel ψ(x1, x2)ψ(y1, y2). Consider the first particle.This system has reduced density matrix given by the integral operator with kernel

(6.13) w(x1, y1) =

∫

ψ(x1, x2)ψ(y1, x2) dx2.

This density matrix defines a quantum mixed state that suffices for predictionsinvolving the first particle alone. It is not a pure state for the first particle, nordoes it characterize a random quantum pure state for the first particle. In fact, thereduced density matrix for the first particle has lost much of the information aboutthe original state of the two particle system.

There is an exception in the case when the original state of the two particlesystem is decomposable, so there is no entanglement. In that case the reduceddensity matrix is a pure quantum state for the first particle. The second particlehas no influence on what is happening with the first particle, and so it may beignored.

A particularly famous example is the composite system obtained by combiningtwo spin 1/2 systems. In Dirac notion the density matrix for the singlet state is

(6.14) ψ0ψ∗0 =

1

2( |↑↓〉〈↑↓| + |↓↑〉〈↓↑| − |↑↓〉〈↓↑| − |↓↑〉〈↑↓| ) .

It is not difficult to rewrite this in matrix notation. For instance, the last term is(6.15)

|↓↑〉〈↑↓|=|↓〉〈↑| ⊗ |↑〉〈↓|=(

01

)

(

1 0)

⊗(

10

)

(

0 1)

=

(

0 01 0

)

⊗(

0 10 0

)

.

46 WILLIAM G. FARIS

The resulting formula is

ψ0ψ∗0 =

1

2

[ (

1 00 0

)

⊗(

0 00 1

)

+

(

0 00 1

)

⊗(

1 00 0

)

−(

0 10 0

)

⊗(

0 01 0

)

−(

0 01 0

)

⊗(

0 10 0

) ]

.(6.16)

After some computation this becomes

ψ0ψ∗0 =

1

4

[ (

1 00 1

)

⊗(

1 00 1

)

−(

1 00 −1

)

⊗(

1 00 −1

)

−(

0 11 0

)

⊗(

0 11 0

)

−(

0 −ii 0

)

⊗(

0 −ii 0

) ]

.(6.17)

The Pauli matrices make an appearance. In brief, this says that

(6.18) ψ0ψ∗0 =

1

4(I ⊗ I − ~σ ⊗ ~σ) .

The singlet state is a rotation invariant pure state.Consider the combined system in the singlet state, and look at the reduced

density matrix for one of the spins. This matrix is

(6.19) W =1

2

(

1 00 1

)

.

This mixed state gives incomplete information about the spin, since it neglects itsintimate relation with the other spin. Another point of view would be to thinkof it as a randomized state. Choose an arbitrary spin direction. Then it is therandomization of the spin up and spin down states in this direction, when the spinup and spin down states are taken with equal probability. This representation as arandom pure state is far from unique.

6.3. Self-adjoint operators as observables? The dynamical role of self-adjoint operators as generators of one-parameter unitary groups is entirely clear.This is given by Stone’s theorem. The role of self-adjoint operators as observablesis comparatively mysterious.

It is not controversial that for every state and every self-adjoint operator Athere is a realization of A as a random variable α defined on a probability space.It is even the case that for every state and every pair of commuting self-adjointoperators A,B there is a realization of A,B as a pair of random variables α, β ona probability space. Suppose that A,D are another pair of commuting self-adjointoperators. Again there is a realization of A,D as a pair of random variables α′, γ′.The problem comes when B,D do not commute. The spectral theorem does notgive a single probability space1 that works for both pairs A,B and A,D.

The random variables α and α′ both correspond to the self-adjoint operatorA. One could imagine that in each concrete realization of the physical system therandom variables α and α′ have equal values. This could be explained if for each

1Such a space exists. Let β and γ′ be conditionally independent given α, and use theprobability distribution for α and the conditional distributions for β given α and for γ′ given α

yielded by quantum mechanics. Since quantum mechanics is silent about the joint distribution of β

and γ′ one is free to do this. (The situation is different with the no-hidden-variables theorems, likeHardy’s, for which the graph corresponding to the relevant quantum mechanical joint probabilitieshas cycles.) This example was communicated by Sheldon Goldstein.


time the experiment is done there is a number ZA associated with the self-adjointoperator A. This quantity would reflect an actual property of the system.

It turns out that there is no such quantity. Self-adjoint operators cannot corre-spond to intrinsic properties of the physical system. There are many impossibilityresults that show this. The most famous of these is based on Bell’s inequality, a re-sult of elementary probability theory. (See the appendix to [12] for an account.) Itshows that if there were properties associated with self-adjoint operators, then thisinequality would be violated. A nice feature of Bell’s analysis is that the physicalsystem is so simple; it is a system of two spin 1/2 particles in the singlet state.

Hardy [7] derived an even more convincing result, in which the contradiction isexhibited on a certain fraction of the individual runs of the experiment, rather thanstatistically. We follow the account in [1]. Again the example involves a systemof two spin 1/2 particles and a certain state of the combined system. Curiouslyenough, the state in Hardy’s example is not the singlet state.

Hardy defines certain spin component operators A,B,C,D. Each has a spec-trum ±1. Say that A,C are associated with the first particle, while B,D areassociated with the second particle. Then the following pairs commute: AB = BA,AD = DA, BC = CB, and CD = DC.

Suppose that self-adjoint operators, considered in isolation, actually correspondto intrinsic properties. Suppose furthermore that calculations involving commut-ing self-adjoint operators proceed in the usual way. For the state considered byHardy and the four self-adjoint operators A,B,C,D there would be correspondingphysical quantities ZA, ZB, ZC , ZD with definite numerical values (depending onthe realization). However Hardy shows that calculations involving only commut-ing self-adjoint operators give the following four results for the joint probabilitiescomputed from the state:

(1) P [ZA = 1, ZB = 1] > 0.(2) P [ZB = 1, ZC = −1] = 0.(3) P [ZD = −1, ZA = 1] = 0.(4) P [ZC = 1, ZD = 1] = 0.

According to 1, in some substantial proportion of repetitions of the experimentthe outcome is such that both ZA and ZB have the value 1. Consider only suchoutcomes. By 2 and 3 both ZC and ZD have the value 1. Then 4 says that thiscannot happen. The supposition that there are such physical quantities has led toa contradiction. They do not exist.

Many physicists are aware of this situation, although perhaps not all realizethat it is fatal for the concept of self-adjoint operator as observable. In fact, theyoften cheerfully acknowledge that for a particular experiment one self-adjoint oper-ator may have an associated physical value, while another self-adjoint operator maynot have such a value at all. Obviously, which is the case must depend on some-thing more than the self-adjoint operator. The self-adjoint operator alone does notdetermine what is observed.

The result of this analysis is that a self-adjoint operators must be combinedwith some other ingredient in order to define an observable. This extra ingredientis the experimental context that defines the measurement. The random physicalquantity, if it is defined at all, depends not only on the self-adjoint operator, but alsoon the experiment. So it is no surprise that the random variable α associated withA in a measurement of A,B may have different values from the random variable

48 WILLIAM G. FARIS

α′ associated with A in measurement of A,D. A possible mechanism for this ispresented in the next subsection.

Nothing in the above discussion proves that there cannot be intrinsic proper-ties of the system, independent of the experimental context. All it shows is thatsuch properties cannot be described exclusively by self-adjoint operators. Thereare variations of non-relativistic quantum mechanics in which a system does haveintrinsic properties. See, for instance, [1] for Bohmian mechanics or the contri-bution by Carlen in [5] for stochastic mechanics. These formulations are oftenoverlooked, but they have considerable conceptual value. In particular, they oftenprovide instant counter-examples to a variety of careless assertions about quantumtheory.

An example of such properties is given by the description of spin in stochasticmechanics. In the model of Dankel the spin variable is allowed to be continuous.When the spinning particle interacts with a magnetic field oriented in a certaindirection, the result is that the spin component becomes oriented in this samedirection and takes on one of two possible values [4]. Furthermore, the spinningparticle is deflected in the corresponding direction. So while spin is an intrinsicproperty of the system, discrete spin only emerges as a result of the interaction. Thismodel gives a vivid illustration of how a self-adjoint operator (the spin componentoperator for the given direction) corresponds to a meaningful physical variable (theactual spin, or the corresponding deflection) only as a result of the introduction ofan extra ingredient (the magnetic field).

The situation is less clear in the case of orthodox quantum mechanics, becauseof a lack of agreement on the interpretation of the theory. It may be that it alsodescribes intrinsic properties in some indirect way, if one can believe the account inthe following subsection. If this were not the case, then it would be a puzzle thatexperiments can have outcomes.

6.4. Self-adjoint operators with measurement as observables. Here isa brief account of one possible interpretation of quantum mechanics. It is influencedby [1], a survey that considers a variety of approaches. The present descriptionattempts to be close to quantum orthodoxy, and, given this constraint, reasonablyprecise and consistent. As in many such accounts, the exposition is marred bythe introduction of undefined terms. In this case, the terms “measurement” and“experiment” are at least partially specified, but the term “macroscopic” is notsubject to serious analysis.

The fundamental idea is that an observable consists of more than a self-adjointoperator; there is also an experimental context. The self-adjoint operator merelyprovides a convenient summary of certain aspects of the situation. There may bemore than one experiment that gives rise to a particular self-adjoint operator. Sothe only kind of measurement involving the self-adjoint operator is when one isperforming the experiment.

We only deal with the case of point spectrum. In this case a self-adjoint operatoracting in the Hilbert space H may be written the form

(6.20) A =∑

j

λjEj


The eigenvalues λj are real numbers (not necessarily distinct). There is an orthog-onal family Hj of closed subspaces of the Hilbert space H whose direct sum is H.The operators Ej denote the orthogonal projections onto the subspaces Hj .

The complete description involves the system together with an environment,forming what one might call the total system. The self-adjoint operator acts on theoriginal system Hilbert space H. The environment Hilbert space H′ is intended todescribe the apparatus used for the experiment. The total system Hilbert space isa tensor product H⊗H′ of original system and environment Hilbert spaces. Thusthe original system is a subsystem of the total system.

An original system vector of the form Ejψ is an eigenvector of the operator Awith eigenvalue λj . Let Φ be the initial state of the environment. Suppose that it ispossible to find a unitary time dynamics U for the total system that sends Ejψ⊗Φto a state

(6.21) U(Ejψ ⊗ Φ) = Rjψ ⊗ Φj .

The environment states Φj are taken to form an orthonormal family. The Rj area family of linear operators on the original system Hilbert space H defined by theabove equation. Since U is unitary, we see that ‖Rjψ‖2 = ‖Ejψ‖2 for each ψ. Inother words, the operators Rj satisfy R∗

jRj = E2j = Ej . The effect of the unitary

dynamics for the total system is to produce a correlation between original systemand the environment in which the original system vectorRjψ goes with environmentvector Φj .

Let ψ be a unit vector representing the initial state of the original system. Sinceψ =

∑

j Ejψ, the unitary dynamics for the total system gives a final state

(6.22) U(ψ ⊗ Φ) =∑

j

Rjψ ⊗ Φj .

The initial state of the original system was a pure state with state vector ψ. Thefinal state of the original system is a mixed state with state vectors given by nor-malizing Rjψ, taken with probability

(6.23) pj = ‖(Rjψ) ⊗ Φj‖2 = ‖Rjψ‖2 = ‖Ejψ‖2 = 〈ψ,Ejψ〉.

This kind of dynamical mapping of a pure state of the original system to a mixedstate of the original system via a unitary map of the total system (of which theoriginal system is a subsystem) is an example of decoherence. In the resultingdescription of the original system all information about the relative phases of thecomponents Ejψ is lost.

It is impossible to map a pure state of the system to a mixed state of the sys-tem with a unitary dynamics of the system alone. In an example of this kind themapping is accomplished with a unitary dynamics for the combined total system.There is no reason to believe that the original system should have a unitary dy-namics obtained from the Schrodinger equation, since there is an interval of timefor which it is not an isolated system.

If we associate the number λj with the random choice of j, then for each realfunction f we have the expectation

(6.24)∑

j

f(λj)pj =∑

j

f(λj)〈ψ,Ejψ〉 = 〈ψ, f(A)ψ〉.

50 WILLIAM G. FARIS

The system operator A together with the pure system state ψ gives the statisticsof λj with the mixed state probabilities pj .

Here is perhaps the simplest example. Consider a spin 1/2 particle. The Hilbertspace is the tensor product of a two-dimensional H with the space of wave functionsH′ = L2(R3). Pick a direction in space, and impose a magnetic field along thisdirection. As the particle passes through the region where this magnetic field acts,it (or rather, its wave function) is deflected in one of two directions, parallel to themagnetic field direction. This deflection is obtained by solving a generalization ofthe Schrodinger equation in which the magnetic field interacts with the spinningparticle, and the solution operator is the unitary operator U . The self-adjointoperator that is relevant to this experiment has spectral projections E± that projectonto the states where the spin in the given direction has one or the other sign. Theunitary operator U maps the initial state ψ ⊗ Φ = (E+ψ + E−ψ) ⊗ Φ to the finalstate R+ψ ⊗ Φ+ + R−ψ ⊗ Φ−. The dynamics is run long enough so that for allpractical purposes the wave functions Φ± have disjoint supports. Thus the finalstate is the sum of two orthogonal states with norms ‖R±ψ‖2 = ‖E±ψ‖2. Theserepresent the probabilities of spin ±1 along the given direction.

Since the time of Bohr it has been customary to interpret the world on theatomic level in terms of experience on the macroscopic scale. In this spirit, con-sider the environment state vectors Φj as being wave functions with macroscopicallydisjoint supports. In this circumstance the correlation with the environment maybe thought of as permanent, for all practical purposes. Therefore no contradictionarises from the supposition that the system state is realized with a particular valueof j. It is then customary to regard λj as the actual experimental value and Rjψ(suitably normalized) as the actual state vector of the system after the measure-ment. This is called reduction or collapse of the state vector. It corresponds to thestandard situation in applications of probability, where it is natural to think thatexperiments that are actually performed have outcomes.

The probabilities of these outcomes are given by the probabilities pj = ‖Ejψ‖2

associated with the self-adjoint operator and the state ψ. However, as we saw inthe previous subsection, the actual outcomes that are obtained must depend onsomething else. It is the association with macroscopically distinct states and theassociated collapse that provides the mechanism.

The Schrodinger equation is deterministic and thus cannot possibly providea random outcome. So collapse is not obtained by solving the Schrodinger equa-tion. It would seem that collapse is a problematic operation from the point ofview of quantum dynamics. On the other hand, collapse does not contradict theSchrodinger equation, since no one expects that the system should obey this equa-tion while it is in interaction with the environment. And the result is that thepure state given by ψ is mapped to another pure state given by Rjψ, suitablynormalized. The idea is that at the beginning and at the end of the experimentthe original system may be regarded as a quantum mechanical system in its ownright, rather than as a subsystem of a larger system. The mapping itself is obtainedby considering the original system as a subsystem of a larger system and invokingdecoherence followed by collapse. From the point of the view of the original sys-tem it appears only as a rather mysterious random process. It is this process thatallows quantum experiments to have physically realized outcomes. For a quantum


mechanical system to have an intrinsic property it would seem that there must bean interaction with the macroscopic world.

Is this all there is to quantum reality? In other words, is it necessary to havethis kind of amplification to the macroscopic level in order to make sense of thetheory? Perhaps the answer is yes, but this leaves many questions unanswered.

We conclude this subsection with a technical comment. The above discussionshowed that an experiment of a certain kind can provide a measurement for a self-adjoint operator. However there is a somewhat more general but still reasonablenotion of experiment that need not provide a such a measurement. Suppose thatthere is a unitary time dynamics U such that for each system vector ψ we have

(6.25) U(ψ ⊗ Φ) =∑

j

Rjψ ⊗ Φj .

Here the Rj are bounded operators, without the restriction that R∗jRj is an or-

thogonal projection. Each individual Oj = R∗jRj is a bounded positive self-

adjoint operator. The unitarity condition implies that for each initial ψ the sum∑

j ‖Rjψ‖2 = ‖ψ‖2. As a consequence, the operators Oj are normalized to sat-

isfy∑

j Oj = I. The same calculation as above shows that the probability of thesupport corresponding to index j is

(6.26) pj = ‖(Rjψ) ⊗ Φj‖2 = ‖Rjψ‖2 = 〈ψ,Ojψ〉.

If we associate the number λj with this event, then there is no difficulty in com-puting

(6.27)∑

j

f(λj)pj =∑

j

f(λj)‖Rjψ‖2 =∑

j

f(λj)〈ψ,Ojψ〉.

So it would seem that the entire discussion of experiment carries over to this moregeneral situation, where there is no reference to a self-adjoint operator A. Thesystem is in a mixed state in which Rjψ occurs with probability pj . The collapseof the state to one of the Rjψ remains just as valid. This form of experiment seemsjust as reasonable as the more restrictive one presented above.

It is still possible to define a self-adjoint operator

(6.28) A =∑

j

λjOj ,

but this representation is far from being a spectral representation. The self-adjointoperator A is useful for expressing the expectation (or mean)

(6.29)∑

j

λjpj =∑

j

λj〈ψ,Ojψ〉 = 〈ψ,Aψ〉,

but it is not relevant to computing the distribution of the λj . In fact the distributionof A is concentrated on its eigenvalues, and the λj are not eigenvalues of A. Thusit is misleading to regard the experiment as providing a measurement for A.

This more general kind of experiment seems just as plausible as what we hadbefore. The result is that certain numbers λj are observed, but they are not numbersassociated with a self-adjoint operator. Perhaps the notion that observables mustcorrespond to self-adjoint operators may be taken with some skepticism.

52 WILLIAM G. FARIS

Acknowledgments

The author thanks Sheldon Goldstein for helpful suggestions. He also thanksthe author of one of the referee reports (apparently a student participant in the“Energy and the Quantum” workshop) for corrections and comments. Finally, he isgrateful to Daniel Ueltschi and Bob Sims for their efforts in organizing the workshopand editing this volume.

References

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[2] William G. Faris, “Perturbations and non-normalizable eigenvectors,” Helvetica Physica Acta44 (1971), 930–936.

[3] William G. Faris, Self-Adjoint Operators (Lecture Notes in Mathematics #433), Springer,Berlin, 1975.

[4] William G. Faris, “Spin correlation in stochastic mechanics,” Found. Phys. 12 (1982), 1–26.[5] William G. Faris (editor), Diffusion, Quantum Theory, and Radically Elementary Mathemat-

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Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

E-mail address: [email protected]

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